Subspace vs Subset: Inheritance of Topology

[tomboi03]

Hey guys...

I'm not sure how I'm suppose to show that if Y is a subspace of X, and A is a subset of Y, then the topology A inherits as a subspace of Y is the same as the topology it inherits as a subspace of X.

I know that a subspace is... T_y = {Y\capU| U \inT}
meaning that its open sets consist of all intersections of open sets of X with Y.
and that a subset is every element of A is also an element of B.

pretty much right? so how do i express this in terms of subset and subspace?

 

[CompuChip]

Let T_Y denote the topology inherited from Y, and T_X the topology inherited from X, i.e.
T_Y = \{ U \cap A | U \text{ is open in } Y \}
and
T_X = \{ V \cap A | V \text{ is open in } X \}

First let's show that T_Y \subseteq T_X. Let U \in T_Y be an open set in the Y-induced topology on A. That means there is some open set U' in Y, such that U = A \cap U'. Can you find a set U'' which is open in X, such that U = A \cap U''? Because that would show that
U \in T_Y \implies U \in T_X
and therefore
T_Y \subseteq T_X.