Difference between revisions of "Compactness in Metric Spaces"

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<p>Furthermore, we said that an open cover (or open covering) is simply a cover that contains only open sets.</p>
 
<p>Furthermore, we said that an open cover (or open covering) is simply a cover that contains only open sets.</p>
 
<p>We also said that a subset <span class="math-inline"><math>\mathcal S \subseteq \mathcal F</math></span> is a subcover/subcovering (or open subcover/subcovering if <span class="math-inline"><math>\mathcal F</math></span> is an open covering) if <span class="math-inline"><math>\mathcal S</math></span> is also a cover of <span class="math-inline"><math>S</math></span>, that is:</p>
 
<p>We also said that a subset <span class="math-inline"><math>\mathcal S \subseteq \mathcal F</math></span> is a subcover/subcovering (or open subcover/subcovering if <span class="math-inline"><math>\mathcal F</math></span> is an open covering) if <span class="math-inline"><math>\mathcal S</math></span> is also a cover of <span class="math-inline"><math>S</math></span>, that is:</p>
−
<span class="equation-number">(2)</span>
+
 
−
<div style="text-align: center;"><math>\begin{align} \quad S \subseteq \bigcup_{A \in \mathcal S} A \quad \text{ where }  \mathcal S \subseteq \mathcal F \end{align}</math></div>
+
<div style="text-align: center;"><math>\begin{align} \quad S \subseteq \bigcup_{A \in \mathcal{S}} A \quad \text{ where }  \mathcal S \subseteq \mathcal F \end{align}</math></div>
 
<p>We can now define the concept of a compact set using the definitions above.</p>
 
<p>We can now define the concept of a compact set using the definitions above.</p>
 
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
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<p>Let <span class="math-inline"><math>n^* = \max \{ n_1, n_2, ..., n_p \}</math></span>. Then due to the nesting of the open covering <span class="math-inline"><math>\mathcal F</math></span>, we see that:</p>
 
<p>Let <span class="math-inline"><math>n^* = \max \{ n_1, n_2, ..., n_p \}</math></span>. Then due to the nesting of the open covering <span class="math-inline"><math>\mathcal F</math></span>, we see that:</p>
 
<div style="text-align: center;"><math>\begin{align} \quad \bigcup_{k=1}^{p} \left ( 0, 1 - \frac{1}{n_p} \right ) = \left ( 0, 1 - \frac{1}{n^*} \right ) \end{align}</math></div>
 
<div style="text-align: center;"><math>\begin{align} \quad \bigcup_{k=1}^{p} \left ( 0, 1 - \frac{1}{n_p} \right ) = \left ( 0, 1 - \frac{1}{n^*} \right ) \end{align}</math></div>
−
<p>But for <span class="math-inline"><math>(0, 1) \subseteq \left ( 0, 1 - \frac{1}{n^*} \right )</math></span> we need <span class="math-inline"><math>1 \leq 1 - \frac{1}{n^*}</math></span>. But <span class="math-inline"><math>n^* \in \mathbb{N}</math></span>, so <span class="math-inline"><math>n^* &gt; 0</math></span> and <span class="math-inline"><math>\frac{1}{n^*} &gt; 0</math></span>, so <span class="math-inline"><math>1 - \frac{1}{n^*} &lt; 1</math></span>. Therefore any finite subset <span class="math-inline"><math>\mathcal F^*</math></span> of <span class="math-inline"><math>\mathcal F</math></span> cannot cover <span class="math-inline"><math>S = (0, 1)</math></span>. Hence, <span class="math-inline"><math>(0, 1)</math></span> is not compact.</p>
+
<p>But for <span class="math-inline"><math>(0, 1) \subseteq \left ( 0, 1 - \frac{1}{n^*} \right )</math></span> we need <span class="math-inline"><math>1 \leq 1 - \frac{1}{n^*}</math></span>. But <span class="math-inline"><math>n^* \in \mathbb{N}</math></span>, so <span class="math-inline"><math>n^* > 0</math></span> and <span class="math-inline"><math>\frac{1}{n^*} > 0</math></span>, so <span class="math-inline"><math>1 - \frac{1}{n^*} < 1</math></span>. Therefore any finite subset <span class="math-inline"><math>\mathcal F^*</math></span> of <span class="math-inline"><math>\mathcal F</math></span> cannot cover <span class="math-inline"><math>S = (0, 1)</math></span>. Hence, <span class="math-inline"><math>(0, 1)</math></span> is not compact.</p>
 +
 
 +
===Boundedness of Compact Sets in a Metric Space===
 +
<p>Recall that if <span class="math-inline"><math>(M, d)</math></span> is a metric space then a subset <span class="math-inline"><math>S \subseteq M</math></span> is said to be compact in <span class="math-inline"><math>M</math></span> if for every open covering of <span class="math-inline"><math>S</math></span> there exists a finite subcovering of <span class="math-inline"><math>S</math></span>.</p>
 +
<p>We will now look at a rather important theorem which will tell us that if <span class="math-inline"><math>S</math></span> is a compact subset of <span class="math-inline"><math>M</math></span> then we can further deduce that <span class="math-inline"><math>S</math></span> is also a bounded subset.</p>
 +
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
<td><strong>Theorem 1:</strong> If <span class="math-inline"><math>(M, d)</math></span> be a metric space and <span class="math-inline"><math>S \subseteq M</math></span> is a compact subset of <span class="math-inline"><math>M</math></span> then <span class="math-inline"><math>S</math></span> is bounded.</td>
 +
</blockquote>
 +
 
 +
<ul>
 +
<li><strong>Proof:</strong> For a fixed <span class="math-inline"><math>x_0 \in S</math></span> and for <span class="math-inline"><math>r > 0</math></span>, consider the ball centered at <span class="math-inline"><math>x_0</math></span> with radius <span class="math-inline"><math>r</math></span>, i.e., <span class="math-inline"><math>B(x_0, r)</math></span>. Let <span class="math-inline"><math>\mathcal F</math></span> denote the collection of balls centered at <span class="math-inline"><math>x_0</math></span> with varying radii <span class="math-inline"><math>r > 0</math></span>:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad \mathcal F = \{ B(x_0, r) : r > 0 \} \end{align}</math></div>
 +
<ul>
 +
<li>It should not be hard to see that <span class="math-inline"><math>\mathcal F</math></span> is an open covering of <span class="math-inline"><math>S</math></span>, since for all <span class="math-inline"><math>s \in S</math></span> we have that <span class="math-inline"><math>d(x_0, s) = r_s > 0</math></span>, so <span class="math-inline"><math>s \in B(x_0, r_s) \in \mathcal F</math></span>.</li>
 +
</ul>
 +
<ul>
 +
<li>Now since <span class="math-inline"><math>S</math></span> is compact and since <span class="math-inline"><math>\mathcal F</math></span> is an open covering of <span class="math-inline"><math>S</math></span>, there exists a finite open subcovering subset <span class="math-inline"><math>\mathcal F^* \subset \mathcal F</math></span> that covers <span class="math-inline"><math>S</math></span>. Since <span class="math-inline"><math>\mathcal F^*</math></span> is finite, we have that:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad \mathcal F^* = \{ B(x_0, r_1), B(x_0, r_2), ..., B(x_0, r_p) \} \end{align}</math></div>
 +
<ul>
 +
<li>And by definition <span class="math-inline"><math>\mathcal F^*</math></span> covers <span class="math-inline"><math>S</math></span> so:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad S \subseteq \bigcup_{k=1}^{p} B(x_0, r_k) \end{align}</math></div>
 +
<ul>
 +
<li>Each of the open balls in the open subcovering <span class="math-inline"><math>\mathcal F^*</math></span> is centered at <span class="math-inline"><math>x_0</math></span> with <span class="math-inline"><math>r_1, r_2, ..., r_p > 0</math></span>. Since the set <span class="math-inline"><math>\{ r_1, r_2, ..., r_p \}</math></span> is a finite set, there exists a maximum value. Let:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad r_{\mathrm{max}} = \max \{ r_1, r_2, ..., r_p \} \end{align}</math></div>
 +
<ul>
 +
<li>Then for all <span class="math-inline"><math>k \in \{ 1, 2, ..., p \}</math></span> we have that <span class="math-inline"><math>B(x_0, r_k) \subseteq B(x_0, r_{\mathrm{max}})</math></span> and therefore:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad S \subseteq \bigcup_{k=1}^{p} B(x_0, r_k) = B(x_0, r_{\mathrm{max}}) \end{align}</math></div>
 +
<ul>
 +
<li>Hence <span class="math-inline"><math>S</math></span> is bounded. <span class="math-inline"><math>\blacksquare</math></span></li>
 +
</ul>
 +
 
 +
===Basic Theorems Regarding Compact Sets in a Metric Space===
 +
<p>Recall that if <span class="math-inline"><math>(M, d)</math></span> is a metric space then a set <span class="math-inline"><math>S \subseteq M</math></span> is said to be compact in <span class="math-inline"><math>M</math></span> if for every open covering of <span class="math-inline"><math>S</math></span> there exists a finite subcovering of <span class="math-inline"><math>S</math></span>.</p>
 +
<p>We will now look at some theorems regarding compact sets in a metric space.</p>
 +
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
<td><strong>Theorem 1:</strong> Let <span class="math-inline"><math>(M, d)</math></span> be a metric space and let <span class="math-inline"><math>S, T \subseteq M</math></span>. Then if <span class="math-inline"><math>S</math></span> is closed and <span class="math-inline"><math>T</math></span> is compact in <span class="math-inline"><math>M</math></span> then <span class="math-inline"><math>S \cap T</math></span> is compact in <span class="math-inline"><math>M</math></span>.</td>
 +
</blockquote>
 +
<ul>
 +
<li><strong>Proof:</strong> Let <span class="math-inline"><math>S</math></span> be closed and let <span class="math-inline"><math>T</math></span> be compact in <span class="math-inline"><math>M</math></span>. Notice that:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad S \cap T \subseteq T \end{align}</math></div>
 +
<ul>
 +
<li>Furthermore, <span class="math-inline"><math>S \cap T</math></span> is closed. This is because <span class="math-inline"><math>S</math></span> is given as closed, and since <span class="math-inline"><math>T</math></span> is compact we know that <span class="math-inline"><math>T</math></span> is closed (and bounded). So the finite intersection <span class="math-inline"><math>S \cap T</math></span> is closed. But any closed subset of a compact set is also compact, so <span class="math-inline"><math>S \cap T</math></span> is compact in <span class="math-inline"><math>M</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
 +
</ul>
 +
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
<td><strong>Theorem 2:</strong> Let <span class="math-inline"><math>(M, d)</math></span> be a metric space and let <span class="math-inline"><math>S_1, S_2, ..., S_n \subseteq M</math></span> be a finite collection of compact sets in <span class="math-inline"><math>M</math></span>. Then <span class="math-inline"><math>\displaystyle{\bigcup_{i=1}^{n} S_i}</math></span> is also compact in <span class="math-inline"><math>M</math></span>.</td>
 +
</blockquote>
 +
<ul>
 +
<li><strong>Proof:</strong> Let <span class="math-inline"><math>S_1, S_2, ..., S_n \subseteq M</math></span> be a finite collection of compact sets in <span class="math-inline"><math>M</math></span>. Consider the union <span class="math-inline"><math>S = \bigcup_{i=1}^{n} S_i</math></span> and let <span class="math-inline"><math>\mathcal F</math></span> be any open covering of <span class="math-inline"><math>S</math></span>, that is:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad S \subseteq \bigcup_{A \in \mathcal F} A \end{align}</math></div>
 +
<ul>
 +
<li>Now since <span class="math-inline"><math>S_i \subseteq S</math></span> for all <span class="math-inline"><math>i \in \{1, 2, ..., n \}</math></span> we see that <span class="math-inline"><math>\mathcal F</math></span> is also an open covering of <span class="math-inline"><math>S</math></span> and so there exists a finite subcollection <span class="math-inline"><math>\mathcal F_i \subseteq \mathcal F</math></span> that also covers <span class="math-inline"><math>S_i</math></span>, i.e.:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad S_i \subseteq \bigcup_{A \in \mathcal F_i} A \end{align}</math></div>
 +
<ul>
 +
<li>Let <span class="math-inline"><math>\mathcal F^* = \bigcup_{i=1}^{n} \mathcal F_i</math></span>. Then <span class="math-inline"><math>\mathcal F^*</math></span> is finite since it is equal to a finite union of finite sets. Furthermore:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad S = \bigcup_{i=1}^{n} S_i \subseteq \bigcup_{i=1}^{n} \left ( \bigcup_{A \in \mathcal F_i} A \right ) = \bigcup_{A \in \mathcal F^*} A \end{align}</math></div>
 +
<ul>
 +
<li>So <span class="math-inline"><math>\mathcal F^* \subseteq \mathcal F</math></span> is a finite open subcovering of <span class="math-inline"><math>S</math></span>. So for all open coverings <span class="math-inline"><math>\mathcal F</math></span> of <span class="math-inline"><math>S</math></span> there exists a finite open subcovering of <span class="math-inline"><math>S</math></span>, so <span class="math-inline"><math>\displaystyle{S = \bigcup_{i=1}^{n} S_i}</math></span> is compact in <span class="math-inline"><math>M</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
 +
</ul>
 +
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
<td><strong>Theorem 3:</strong> Let <span class="math-inline"><math>(M, d)</math></span> be a metric space and let <span class="math-inline"><math>\mathcal C</math></span> be an arbitrary collection of compact sets in <span class="math-inline"><math>M</math></span>. Then <span class="math-inline"><math>\displaystyle{\bigcap_{C \in \mathcal C} C}</math></span> is also compact in <span class="math-inline"><math>M</math></span>.</td>
 +
</blockquote>
 +
<ul>
 +
<li><strong>Proof:</strong> Let <span class="math-inline"><math>\mathcal C</math></span> be an arbitrary collection of compact sets in <span class="math-inline"><math>M</math></span>. Notice that for all <span class="math-inline"><math>C \in \mathcal C</math></span> that:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad \bigcap_{C \in \mathcal C} C \subseteq C \end{align}</math></div>
 +
<ul>
 +
<li>Furthermore, since each <span class="math-inline"><math>C \in \mathcal C</math></span> is compact, then each <span class="math-inline"><math>C</math></span> is closed (and bounded). An arbitrary intersection of closed sets is closed, and so <span class="math-inline"><math>\displaystyle{\bigcap_{C \in \mathcal C}}</math></span> is a closed subset of the compact set <span class="math-inline"><math>C</math></span>. Therefore by the theorem referenced earlier, <span class="math-inline"><math>\displaystyle{\bigcap_{C \in \mathcal C} C}</math></span> is compact in <span class="math-inline"><math>M</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
 +
</ul>
 +
 
 +
==Licensing==
 +
Content obtained and/or adapted from:
 +
* [http://mathonline.wikidot.com/compact-sets-in-a-metric-space Compact Sets in a Metric Space, mathonline.wikidot.com] under a CC BY-SA license
 +
* [http://mathonline.wikidot.com/boundedness-of-compact-sets-in-a-metric-space Boundedness of Compact Sets in a Metric Space, mathonline.wikidot.com] under a CC BY-SA license
 +
* [http://mathonline.wikidot.com/basic-theorems-regarding-compact-sets-in-a-metric-space Basic Theorems Regarding Compact Sets in a Metric Space, mathonline.wikidot.com] under a CC BY-SA license

Latest revision as of 13:56, 9 November 2021

Compact Sets in a Metric Space

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} is a metric space and then a cover or covering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a collection of subsets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} such that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad S \subseteq \bigcup_{A \in \mathcal F} A \end{align}}

Furthermore, we said that an open cover (or open covering) is simply a cover that contains only open sets.

We also said that a subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal S \subseteq \mathcal F} is a subcover/subcovering (or open subcover/subcovering if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} is an open covering) if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal S} is also a cover of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , that is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad S \subseteq \bigcup_{A \in \mathcal{S}} A \quad \text{ where } \mathcal S \subseteq \mathcal F \end{align}}

We can now define the concept of a compact set using the definitions above.

Definition: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} be a metric space. The subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq M} is said to be Compact if every open covering Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} has a finite subcovering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} .

In general, it may be more difficult to show that a subset of a metric space is compact than to show a subset of a metric space is not compact. So, let's look at an example of a subset of a metric space that is not compact.

Consider the metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{R}, d)} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is the Euclidean metric and consider the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = [0, 1) \subseteq \mathbb{R}} . We claim that this set is not compact. To show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is not compact, we need to find an open covering Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} that does not have a finite subcovering. Consider the following open covering:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mathcal F = \left \{ \left( (0, 1 - \frac{1}{n} \right ) : n \in \mathbb{Z}, n \geq 1 \right \} = \left \{ \left ( 0, \frac{1}{2}, \right ), \left ( 0, \frac{2}{3} \right ) , \left ( 0, \frac{3}{4} \right ) , ... \right \} \end{align}}

Clearly Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} is an infinite subcovering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, 1)} and furthermore:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (0, 1) \subseteq \bigcup_{k=2}^{\infty} \left ( 0, 1 - \frac{1}{k} \right ) \end{align}}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^*} be a finite subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} containing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} elements. Then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mathcal F^* = \left \{ \left (0, 1 - \frac{1}{n_1}\right ), \left (0, 1 - \frac{1}{n_2}\right ), ..., \left (0, 1 - \frac{1}{n_p}\right ) \right \} \end{align}}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^* = \max \{ n_1, n_2, ..., n_p \}} . Then due to the nesting of the open covering Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} , we see that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \bigcup_{k=1}^{p} \left ( 0, 1 - \frac{1}{n_p} \right ) = \left ( 0, 1 - \frac{1}{n^*} \right ) \end{align}}

But for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, 1) \subseteq \left ( 0, 1 - \frac{1}{n^*} \right )} we need Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \leq 1 - \frac{1}{n^*}} . But Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^* \in \mathbb{N}} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^* > 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{n^*} > 0} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 - \frac{1}{n^*} < 1} . Therefore any finite subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^*} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} cannot cover Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = (0, 1)} . Hence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, 1)} is not compact.

Boundedness of Compact Sets in a Metric Space

Recall that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} is a metric space then a subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq M} is said to be compact in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} if for every open covering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} there exists a finite subcovering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} .

We will now look at a rather important theorem which will tell us that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a compact subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} then we can further deduce that is also a bounded subset.

Theorem 1: If be a metric space and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq M} is a compact subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is bounded.

  • Proof: For a fixed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0 \in S} and for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} , consider the ball centered at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} with radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} , i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(x_0, r)} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} denote the collection of balls centered at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} with varying radii Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} :
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mathcal F = \{ B(x_0, r) : r > 0 \} \end{align}}
  • It should not be hard to see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} is an open covering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , since for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \in S} we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x_0, s) = r_s > 0} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \in B(x_0, r_s) \in \mathcal F} .
  • Now since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is compact and since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} is an open covering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , there exists a finite open subcovering subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^* \subset \mathcal F} that covers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^*} is finite, we have that:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mathcal F^* = \{ B(x_0, r_1), B(x_0, r_2), ..., B(x_0, r_p) \} \end{align}}
  • And by definition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^*} covers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} so:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad S \subseteq \bigcup_{k=1}^{p} B(x_0, r_k) \end{align}}
  • Each of the open balls in the open subcovering Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^*} is centered at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_1, r_2, ..., r_p > 0} . Since the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ r_1, r_2, ..., r_p \}} is a finite set, there exists a maximum value. Let:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad r_{\mathrm{max}} = \max \{ r_1, r_2, ..., r_p \} \end{align}}
  • Then for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \in \{ 1, 2, ..., p \}} we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(x_0, r_k) \subseteq B(x_0, r_{\mathrm{max}})} and therefore:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad S \subseteq \bigcup_{k=1}^{p} B(x_0, r_k) = B(x_0, r_{\mathrm{max}}) \end{align}}
  • Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is bounded. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

Basic Theorems Regarding Compact Sets in a Metric Space

Recall that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} is a metric space then a set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq M} is said to be compact in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} if for every open covering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} there exists a finite subcovering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} .

We will now look at some theorems regarding compact sets in a metric space.

Theorem 1: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} be a metric space and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S, T \subseteq M} . Then if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is closed and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is compact in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \cap T} is compact in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} .

  • Proof: Let be closed and let be compact in . Notice that:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad S \cap T \subseteq T \end{align}}
  • Furthermore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \cap T} is closed. This is because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is given as closed, and since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is compact we know that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is closed (and bounded). So the finite intersection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \cap T} is closed. But any closed subset of a compact set is also compact, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \cap T} is compact in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

Theorem 2: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} be a metric space and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1, S_2, ..., S_n \subseteq M} be a finite collection of compact sets in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\bigcup_{i=1}^{n} S_i}} is also compact in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} .

  • Proof: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1, S_2, ..., S_n \subseteq M} be a finite collection of compact sets in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Consider the union Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = \bigcup_{i=1}^{n} S_i} and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} be any open covering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , that is:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad S \subseteq \bigcup_{A \in \mathcal F} A \end{align}}
  • Now since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_i \subseteq S} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in \{1, 2, ..., n \}} we see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} is also an open covering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and so there exists a finite subcollection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F_i \subseteq \mathcal F} that also covers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_i} , i.e.:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad S_i \subseteq \bigcup_{A \in \mathcal F_i} A \end{align}}
  • Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^* = \bigcup_{i=1}^{n} \mathcal F_i} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^*} is finite since it is equal to a finite union of finite sets. Furthermore:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad S = \bigcup_{i=1}^{n} S_i \subseteq \bigcup_{i=1}^{n} \left ( \bigcup_{A \in \mathcal F_i} A \right ) = \bigcup_{A \in \mathcal F^*} A \end{align}}
  • So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^* \subseteq \mathcal F} is a finite open subcovering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} . So for all open coverings Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} there exists a finite open subcovering of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{S = \bigcup_{i=1}^{n} S_i}} is compact in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

Theorem 3: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} be a metric space and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal C} be an arbitrary collection of compact sets in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\bigcap_{C \in \mathcal C} C}} is also compact in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} .

  • Proof: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal C} be an arbitrary collection of compact sets in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Notice that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \in \mathcal C} that:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \bigcap_{C \in \mathcal C} C \subseteq C \end{align}}
  • Furthermore, since each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \in \mathcal C} is compact, then each is closed (and bounded). An arbitrary intersection of closed sets is closed, and so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\bigcap_{C \in \mathcal C}}} is a closed subset of the compact set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} . Therefore by the theorem referenced earlier, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\bigcap_{C \in \mathcal C} C}} is compact in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

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