Difference between revisions of "Compactness in Metric Spaces"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
Line 54: Line 54:
 
<li>Hence <span class="math-inline"><math>S</math></span> is bounded. <span class="math-inline"><math>\blacksquare</math></span></li>
 
<li>Hence <span class="math-inline"><math>S</math></span> is bounded. <span class="math-inline"><math>\blacksquare</math></span></li>
 
</ul>
 
</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 as we proved on the <a href="/closed-subsets-of-compact-sets-in-metric-spaces">Closed Subsets of Compact Sets in Metric Spaces</a> page, 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==
 
==Licensing==
Line 59: Line 102:
 
* [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/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/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

Revision as of 13:55, 9 November 2021

Compact Sets in a Metric Space

If is a metric space and then a cover or covering of is a collection of subsets in such that:

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 is a subcover/subcovering (or open subcover/subcovering if is an open covering) if is also a cover of , that is:

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

Definition: Let 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 .

  • Proof: Let be closed 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 T} 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} . 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 as we proved on the <a href="/closed-subsets-of-compact-sets-in-metric-spaces">Closed Subsets of Compact Sets in Metric Spaces</a> page, 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 is compact, then 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} 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}


Licensing

Content obtained and/or adapted from:

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Compactness_in_Metric_Spaces&oldid=3676"