Compactness in Metric Spaces

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 $S$ that does not have a finite subcovering. Consider the following open covering:

${\begin{aligned}\quad {\mathcal {F}}=\left\{\left((0,1-{\frac {1}{n}}\right):n\in \mathbb {Z} ,n\geq 1\right\}=\left\{\left(0,{\frac {1}{2}},\right),\left(0,{\frac {2}{3}}\right),\left(0,{\frac {3}{4}}\right),...\right\}\end{aligned}}$

Clearly ${\mathcal {F}}$ is an infinite subcovering of $(0,1)$ and furthermore:

${\begin{aligned}\quad (0,1)\subseteq \bigcup _{k=2}^{\infty }\left(0,1-{\frac {1}{k}}\right)\end{aligned}}$

Let ${\mathcal {F}}^{*}$ be a finite subset of ${\mathcal {F}}$ containing $p$ elements. Then:

${\begin{aligned}\quad {\mathcal {F}}^{*}=\left\{\left(0,1-{\frac {1}{n_{1}}}\right),\left(0,1-{\frac {1}{n_{2}}}\right),...,\left(0,1-{\frac {1}{n_{p}}}\right)\right\}\end{aligned}}$

Let $n^{*}=\max\{n_{1},n_{2},...,n_{p}\}$. Then due to the nesting of the open covering ${\mathcal {F}}$, we see that:

${\begin{aligned}\quad \bigcup _{k=1}^{p}\left(0,1-{\frac {1}{n_{p}}}\right)=\left(0,1-{\frac {1}{n^{*}}}\right)\end{aligned}}$

But for $(0,1)\subseteq \left(0,1-{\frac {1}{n^{*}}}\right)$ we need $1\leq 1-{\frac {1}{n^{*}}}$. But $n^{*}\in \mathbb {N}$, so $n^{*}>0$ and ${\frac {1}{n^{*}}}>0$, so $1-{\frac {1}{n^{*}}}<1$. Therefore any finite subset ${\mathcal {F}}^{*}$ of ${\mathcal {F}}$ cannot cover $S=(0,1)$. Hence, $(0,1)$ is not compact.

Boundedness of Compact Sets in a Metric Space

Recall that if $(M,d)$ is a metric space then a subset $S\subseteq M$ is said to be compact in $M$ if for every open covering of $S$ there exists a finite subcovering of $S$.

We will now look at a rather important theorem which will tell us that if $S$ is a compact subset of $M$ then we can further deduce that $S$ is also a bounded subset.

Theorem 1: If $(M,d)$ be a metric space and $S\subseteq M$ is a compact subset of $M$ then $S$ is bounded.

• Proof: For a fixed $x_{0}\in S$ and for $r>0$, consider the ball centered at $x_{0}$ with radius $r$, i.e., $B(x_{0},r)$. Let ${\mathcal {F}}$ denote the collection of balls centered at $x_{0}$ with varying radii $r>0$:

${\begin{aligned}\quad {\mathcal {F}}=\{B(x_{0},r):r>0\}\end{aligned}}$

• It should not be hard to see that ${\mathcal {F}}$ is an open covering of $S$, since for all $s\in S$ we have that $d(x_{0},s)=r_{s}>0$, so $s\in B(x_{0},r_{s})\in {\mathcal {F}}$.

• Now since $S$ is compact and since ${\mathcal {F}}$ is an open covering of $S$, there exists a finite open subcovering subset ${\mathcal {F}}^{*}\subset {\mathcal {F}}$ that covers $S$. Since ${\mathcal {F}}^{*}$ is finite, we have that:

${\begin{aligned}\quad {\mathcal {F}}^{*}=\{B(x_{0},r_{1}),B(x_{0},r_{2}),...,B(x_{0},r_{p})\}\end{aligned}}$

• And by definition ${\mathcal {F}}^{*}$ covers $S$ so:

${\begin{aligned}\quad S\subseteq \bigcup _{k=1}^{p}B(x_{0},r_{k})\end{aligned}}$

• Each of the open balls in the open subcovering ${\mathcal {F}}^{*}$ is centered at $x_{0}$ with $r_{1},r_{2},...,r_{p}>0$. Since the set $\{r_{1},r_{2},...,r_{p}\}$ is a finite set, there exists a maximum value. Let:

${\begin{aligned}\quad r_{\mathrm {max} }=\max\{r_{1},r_{2},...,r_{p}\}\end{aligned}}$

• Then for all $k\in \{1,2,...,p\}$ we have that $B(x_{0},r_{k})\subseteq B(x_{0},r_{\mathrm {max} })$ and therefore:

${\begin{aligned}\quad S\subseteq \bigcup _{k=1}^{p}B(x_{0},r_{k})=B(x_{0},r_{\mathrm {max} })\end{aligned}}$

• Hence $S$ is bounded. $\blacksquare$

Basic Theorems Regarding Compact Sets in a Metric Space

Recall that if $(M,d)$ is a metric space then a set $S\subseteq M$ is said to be compact in $M$ if for every open covering of $S$ there exists a finite subcovering of $S$.

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

Theorem 1: Let $(M,d)$ be a metric space and let $S,T\subseteq M$. Then if $S$ is closed and $T$ is compact in $M$ then $S\cap T$ is compact in $M$.

• Proof: Let $S$ be closed and let $T$ be compact in $M$. 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:

${\begin{aligned}\quad \bigcap _{C\in {\mathcal {C}}}C\subseteq C\end{aligned}}$

• 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 Failed to parse (MathML with SVG or PNG fallback (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:

• Compact Sets in a Metric Space, mathonline.wikidot.com under a CC BY-SA license
• Boundedness of Compact Sets in a Metric Space, mathonline.wikidot.com under a CC BY-SA license
• Basic Theorems Regarding Compact Sets in a Metric Space, mathonline.wikidot.com under a CC BY-SA license