Could you show me how to do 3.30 in detail? Transcribed Image Text: Theorem 3.30. Let (X,J) be a topological space, and let (Y, Fy) be a subspace. If B is a
basis for T, then By = {Bn Y|B € B} is a basis for Fy.
Definition. Let (X,T) be a topological space. For Y C X, the collection
Ty = {U | U = Vn Y for some V E I}
is a topology on Y called the subspace topology. It is also called the relative topology
on Y inherited from X. The space (Y, Ty) is called a (topological) subspace of X. If
U E Ty we say U is open in Y.
Theorem 3.25. Let (X,T) be a topological space and Y cX. Then the collection of sets
Ty is in fact a topology on Y.
Theorem 3.28. Let (Y, Fý) be a subspace of (X,J). A subset C c Y is closed in (Y, Fy)
if and only if there is a set D C X, closed in (X,T), such that C = Dn Y.
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