tags:theorem

Topological Space Induced by the Metric Space

Context

Statements

Theorem (@Eck23, Theorem 1.2, @Men90, Theorem 6.4). Let $(X,d)$ be a metric space. Then, $(X,d)$ is a topological space, i.e., the metric open sets in $X$ satisfy the following properties:

• $X$ is open and $\varnothing$ is open.
• The union of any collection (finite or infinite) of open sets is open.
• The intersection of any finite collection of open sets is open.

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