## Proof that The Closed Sets in a Subspace of a Topological Space X are the Intersections of the Subspace With the Closed Sets in X

Theorem

Letbe a topological space and letbe a subspace. A setis closed inif and only ifwhereis closed in

Proof

Supposeis closed inthenwhereis open in

Thenwhere

Hence

Sinceis closed in

Supposewhereis closed in

Then

whereis open inandis open inHenceis closed in