How is a metric space a topological space?

Question

I learned about metric spaces and topological spaces but I don't see how they correlate.

How does a metric space follow the properties of a topological space.

Answer 1

You call a set open if it contains a ball of positive diameter around each of its points. It can be (easily) shown this follows the usual axioms of a topological space.

It can be partially reversed (not so easy): Every $T_3$ space with a countable basis is metrizable.

Answer 2

Let $(X,d)$ be a metric space. Then, define $B_\epsilon(x) = \{y \in X| d(x,y) < \epsilon\}$ to be the ball with radius $\epsilon$ around $x$. You can now define a topology on $X$ as follows:

A subset $U \subset X$ is open, if, for all $x \in U$, there is $\epsilon > 0$ such that $B_\epsilon(x)$ is contained in $U$. One easily checks that the so defined collection of open sets defines a topology.

Note that, if $X = \mathbb{R}^n$ and $d$ is the usual Euclidean distance, this coincides with the usual Euclidean topology on $\mathbb{R}^n$.

Answer 3

The classical answer is that given any metric space, using open balls you can define a topology (where a set is open if every point in it brings with it an open ball contained in the set). In fact, metric spaces and topological spaces are much more closely related than that, if one interprets metric spaces in a suitable generalized form. Using Flagg's notion of value quantale (a certain partially ordered sets with just the right structure to define metric spaces valued in it), it is still the case that every such generalized metric space induces a topology, but now in fact any topology is metrizable, i.e., there exists a value quantale tailored for it and a metric structure inducing the given topology. This correspondence can be made to explicit give an equivalence between the category of topological spaces and the category of generalized metric spaces.

Answer 4

You can equip any metric space with a valid topology:

Topologies can be defined by designating a generating basis. In the case of a metric space, $X$, with a metric, $d$, we can consider our generating neighborhoods to be any set of the form $\{y\in X|d(y,x)<\epsilon\}$ given some center point $x\in X$. This is usually called the "metric topology".