Induced Topology of a Metric

Statements

Proposition. [^1] Let $(X,d)$ a metric space, if for every $y\in X$, there exists $\epsilon >0$, such that the open ball

$$B_{<\epsilon }(y)=\{x\in X|d(y,x)<\epsilon \}$$

satisfies $B_{<\epsilon }(y)\subset X$, then $X$ is an open set and there exists a collection of open balls that forms a topology $\tau$ on $X$ and the triple $(X,d,\tau )$ forms a topological space.

Proof. Let $\tau =\{B_{<i}(y){\}}_{i\in \mathbb{N}}$ be a sequence of open balls. Under the hypothesis of the proposition, $X$ and $\varnothing$ belong to the sequence of open balls. The other two properties follows from the definition of the topology. Thus, $X$ is a topological open set.

References

[1] https://math.stackexchange.com/questions/1409687/what-is-the-topology-induced-by-a-metric/1409698