# Subspace topologies

• September 24th 2011, 09:22 PM
dwsmith
Subspace topologies
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.

Since Y is a subspace, $T_Y=\{Y\cap U: \ \text{U is open in X}\}$

Not sure what to do next.
• September 24th 2011, 09:39 PM
FernandoRevilla
Re: Subspace topologies
Hint As $A\subset Y$ then $A\cap (Y\cap U)=(A\cap Y)\cap U=A\cap U$