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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 E B} is a basis for Ty.

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 E B} is a basis for Ty.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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. %3D
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. %3D
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