Metric Topology

Last updated Nov 1, 2022

Let $(M, d)$ be a Metric Space. The Metric Topology on $M$ is the topology generated by the basis

$$\mathcal{B} = {B_{\epsilon}(x) \subset M: x \in M, \epsilon > 0}$$ where $B_{\epsilon}(x) = {y \in X : d(y, x) < \epsilon}$ (Open Balls on $M$).

1. We should verify that $\mathcal{B}$ is indeed a Topological Basis of our Topological Space:

   Proof: First observe that $\forall x \in M$, $x \in B_{1}(x) \in \mathcal{B}$. Next suppose $B_{\epsilon}(x), B_{\delta}(y) \in \mathcal{B}$ for $\epsilon, \delta > 0$ and $x, y \in M$. Suppose $z \in B_{\epsilon}(x) \cap B_{\delta}(y)$. Because Finite Intersections of Open Balls contain Open Balls about each point, we see there exists $\zeta > 0$ so that $B_{\zeta}(z) \subset B_{\epsilon}(x) \cap B_{\delta}(y)$. Therefore $\mathcal{B}$ is a valid Topological Basis. $\blacksquare$

2. A Set is Open in the Metric Topology iff it contains a Ball around each Point

3. When we talk about a Metric Space, we implicitly endow it with the Metric Topology