A metric space is a topological space ↩

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created: 2021-12-15 11:47:23
modified: 2022-01-11 15:18:53

Let

be a metric space and

be the collection of all open subsets of

. We can show that

and

satisfies the axioms of topological spaces:

T1

Proof

This follows from noting that the empty set is open since has no elements and obviously is open.

T2

The union of any members of is a member if

Proof

Any point of the union of open sets belongs to (at least) one of the sets, call it , and contains a ball about since is open. Then , by the definition of a union. QED

T3

The intersection of finitely many members of is a member of >)

Proof

If is any point of the intersection of open sets , then each contains a ball about and a smallest of these balls is contained in that intersection