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$$\cap$$U| U $$\in$$T}
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
$$T_Y \subseteq T_X$$.