Difference between revisions of "Compactness in Metric Spaces"

Revision as of 15:53, 8 November 2021

Compact Sets in a Metric Space

If $(M,d)$ is a metric space and $S\subseteq M$ then a cover or covering of $S$ is a collection of subsets ${\mathcal {F}}$ in $M$ such that:

{\begin{aligned}\quad S\subseteq \bigcup _{A\in {\mathcal {F}}}A\end{aligned}}

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 ${\mathcal {S}}\subseteq {\mathcal {F}}$ is a subcover/subcovering (or open subcover/subcovering if ${\mathcal {F}}$ is an open covering) if ${\mathcal {S}}$ is also a cover of $S$ , that is:

{\begin{aligned}\quad S\subseteq \bigcup _{A\in {\mathcal {S}}}A\quad {\text{ where }}{\mathcal {S}}\subseteq {\mathcal {F}}\end{aligned}}

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

Definition: Let $(M,d)$ be a metric space. The subset $S\subseteq M$ is said to be Compact if every open covering ${\mathcal {F}}$ of $S$ has a finite subcovering of $S$ .

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

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

Licensing

Content obtained and/or adapted from:

Compact Sets in a Metric Space, mathonline.wikidot.com under a CC BY-SA license

@@ Line 20: / Line 20: @@
 <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^* > 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>
+<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/boundedness-of-compact-sets-in-a-metric-space/Screen%20Shot%202015-10-05%20at%209.56.39%20PM.png" alt="Screen%20Shot%202015-10-05%20at%209.56.39%20PM.png" class="image" /></div>
+<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 &gt; 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 &gt; 0</math></span>:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad \mathcal F = \{ B(x_0, r) : r &gt; 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 &gt; 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 &gt; 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>
 ==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

Difference between revisions of "Compactness in Metric Spaces"

Revision as of 15:53, 8 November 2021

Compact Sets in a Metric Space

Boundedness of Compact Sets in a Metric Space

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