5.1. Metric spaces

Topology plays in important role in one of the two learning objectives for this course: Abstraction. Concretely, there will be some technical definitions, such as the one below, that I will swiftly “forget” and leave behind in favor of more general, much more powerful ideas. The most important takeaway of this chapter is:

Topology allows to talk about closeness without having to talk about distance.

Definition.

A metric space is a structure $(X,d)$ where $X$ is any set and the function $d:X\times X\to \mathbb{R}$, called the metric, has the following properties for any $x,y,z\in X$.

1. $d(x,x)=0$;
2. $d(x,y)>0$ whenever $x\ne y$ (Positivity);
3. $d(x,y)=d(y,x)$ (Symmetry); and
4. $d(x,z)\le d(x,y)+d(y,z)$ (Triangle inequality).

Example: graphs

If $G$ is a connected graph, define $d(u,v)$ as the smallest length of a path with endpoints $u$ and $v$. $(G,d)$ is then a metric space.

Exercise.

If $\phi :G\to H$ is a graph homomorphism, then for all $x,y\in V(G)$, $d(\phi (x),\phi (y))\le d(x,y)$.

If $\phi$ is an isomorphism, then the above inequality becomes an equality.

For a more concrete example, define in any component $B$ of the hypercube $B_{\infty }$, where $s$ and $t$ are countably infinite binary sequences, $h(s,t)$ as the number of coordinates that $s$ and $t$ differ in. That is,

$$h(s,t):=|\{n\in \mathbb{N}:s(n)\ne t(n)\}|.$$

$(B,h)$ is then a metric space. This particular $h$ is known as the Hamming distance, and it plays an important role in error-correcting codes.

Example: the reals

Another example: ${\mathbb{R}}^n$ with the Euclidean metric given by

$$d(\vec{x},\vec{y}):=\sqrt{\sum _{i=1}^n(x_i-y_i{)}^2}.$$

When $n=1$, this is just $d(x,y)=|x-y|$.

For a different example,

$$d^{\prime }(\vec{x},\vec{y}):=\sum _{i=1}^n|x_i-y_i|$$

is sometimes referred to as the taxicab metric. To see why, draw ${\mathbb{R}}^2$ as a grid and imagine roads connecting the lattice points.

Open and closed sets

From now on, let $(X,d)$ be an abstract metric space. Since my course only deals with a very short introduction, for the most part you can safely assume that $X=\mathbb{R}$ with the Euclidean metric.

Definition of open.

The open ball centered at $x$ and of radius $r>0$ is the set $B(x,r):=\{y\in X:d(x,y)<r\}$.

A set $A\subseteq X$ is open if for every $x\in A$, there is some $r>0$ such that $B(x,r)\subseteq A$.

The collection of all open sets is denoted by $\tau (X)$ and is called the topology of $X$.

An immediate remark is that open balls are open.

Examples.

1. In $X=\mathbb{R}$, open balls are open intervals of the form $(x-r,x+r)$.
2. Furthermore, show that every open interval is open.
3. $(-\infty ,-1)\cup (1,\infty )$ is an open set that is not an interval.
4. Draw an open ball centered at the origin of radius 1 in $X={\mathbb{R}}^n$ for $n=2,3$.
5. What are the open balls in a connected graph?
6. Now draw open balls with the other metrics mentioned above. How are they different?

Definition of closed.

Given $x\in X$ and $A\subseteq X$, I say $x$ is a point of closure of $A$ if for all $r>0$, $B(x,r)\cap A\ne \varnothing$. The set of all these points is called the closure of $A$ and is denoted by $\overline{A}$.

I say $A$ is closed if $\overline{A}\subseteq A$.

It is obvious that $A\subseteq \overline{A}$.

Examples.

1. The interval $(0,1)$ does not contain $0$ or $1$, but these are points of closure.
2. What are the points of closure of all of the sets mentioned in the previous example?
3. What are the points of closure of the set $\{1/n:n>1\}$?
4. $\overline{[0,1]}=[0,1]$, so this set is closed.
5. $\overline{Q}=\mathbb{R}$ (Density).

Exercise.

Prove that $\overline{\overline{A}}=\overline{A}$ (Idempotency). In particular, closures are always closed. Put in different words, a set is closed if and only if it is equal to some closure.

The following theorem is the actual definition of an abstract topology, which I will not cover in this course.

Theorem ("Metric spaces are topological spaces").

1. $X$ and $\varnothing$ are open.
2. The union of open sets is open.
3. The intersection of finitely many closed sets is closed.

From now on, I will denote the collection of all open subsets of $X$ by $\tau (X)$. This set is typically called the topology of $X$.

An important connection between open and closed sets is the following.

Proposition.

A set $A$ is open if and only if ${\mathbb{R}}^n\setminus A$ is closed.

Proved in PS5-1.

Another easy-to-prove property of metric spaces is seen below.

Theorem ("Metric spaces have the Hausdorff separation property").

For any distinct $x,y\in X$, there are disjoint open balls $B$ and $B^{\prime }$ such that $x\in B$ and $y\in B^{\prime }$.

Proof:

Since $x\ne y$, $r:=\frac{1}{2}d(x,y)>0$. Thus $B(x,r)$ and $B(y,r)$ have the desired property.

Interior and closure operators

Definition of interior.

Let $A\subseteq X$. Then $\mathrm{int}A$ is the union of all open sets contained in $A$.

Basic properties of interior and closure.

1. $\mathrm{int}A\subseteq A\subseteq \overline{A}$ and the three sets are equal if and only if $A\in \{\varnothing ,X\}$.
2. If $A\subseteq B$, then $\overline{A}\subseteq \overline{B}$ and $\mathrm{int}A\subseteq \mathrm{int}B$. (Monotonicity)
3. $\mathrm{int}A$ is the union of all open balls contained in $A$.
4. $A\in \tau$ if and only if $A\subseteq \mathrm{int}A$. In particular, the interior is always open and the closure is always closed.
5. $\mathrm{int}A\cap \mathrm{int}B=\mathrm{int}(A\cap B)$ and $\overline{A}\cup \overline{B}=\overline{A\cup B}$.
6. $\mathrm{int}(X\setminus A)=X\setminus \overline{A}$ and $\overline{X\setminus A}=X\setminus \mathrm{int}A$.

Exercises.

1. Prove the above proposition.
2. Find counterexamples for the converse implications of (2).
3. Find counterexamples for (5) interchanging union and intersection.
4. Prove that if $A$ is closed and $B$ has empty interior, then $\mathrm{int}(A\cup B)=\mathrm{int}A$.
5. Give a counterexample of the above after removing the word closed.

Sequences

Recall that a sequence in $X$ is just a function $f:\mathbb{N}\to X$, which I denote by its indices $x_n=f(n)$ when convenient.

Notation/Definition of limit.

Let ${\lim }_{n\to \infty }{x}_n=x$ abbreviate the following statement.

$$\forall r>0\exists N\in \mathbb{N}\forall n\ge N{x}_n\in B(x,r)$$

Example.

I will prove that ${\lim }_{n\to \infty }\frac{1}{n}=0$. Let $r>0$, then, by the Archimedean property, there is an $N\in \mathbb{N}$ such that $N>\frac{1}{r}$. But then $\frac{1}{N}<r$, so obviously $\frac{1}{n}\in B(0,r)$ for all $n\ge N$.

Exercise.

If $A$ is a closed set and $\{x_n{\}}_n\subseteq A$ has a limit $x$, then $x\in A$.

Theorem ("In metric spaces, sequential closure and closure are the same").

For any set $A\subseteq X$, $\overline{A}=\{x\in X:\exists \{x_n{\}}_n\subseteq A{\lim }_{n\to \infty }{x}_n=x\}$.

Proof:

$\boxed{\subseteq }$ Let $x\in \overline{A}$. For every natural $n$, pick $x_n\in A\cap B(x,\frac{1}{n+1})$, then clearly $\{x_n{\}}_n\subseteq A$.

Exercise.

Verify that ${\lim }_{n\to \infty }{x}_n=x$.

Therefore, $x$ is a member of the set on the right-hand side.

$\boxed{\supseteq }$ Suppose that ${\lim }_{n\to \infty }{x}_n=x$ for some $\{x_n{\}}_n\subseteq A$. Let $r>0$. Then, for some large natural $N$, $x_N\in B(x,r)$. Clearly this implies that $B(x,r)\cap A\ne \varnothing$ and so $x\in \overline{A}$.