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
Let
be a topological space and let
be a subspace. A set
is closed in
if and only if
where
is closed in
Proof
Supposeis closed inthen
where
is open in
Then
where
Hence
Since
is closed in
Supposewhereis closed in
Then
where
is open inand
is open in
Hence
is closed in