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 $Failed to parse (syntax error): {\displaystyle r > 0}$ , 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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0}$ :

{\begin{aligned}\quad {\mathcal {F}}=\{B(x_{0},r):r&gt;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 $Failed to parse (syntax error): {\displaystyle d(x_0, s) = r_s > 0}$ , 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 $Failed to parse (syntax error): {\displaystyle r_1, r_2, ..., r_p > 0}$ . Since the set $\{r_{1},r_{2},...,r_{p}\}$ is a finite set, there exists a maximum value. Let: