# Subspace topology

• February 28th 2009, 10:22 AM
horowitz
Subspace topology
Let X be a topological space, and let Y subset of X have the subspace topology.
(a) If A is open in Y, and Y is open in X, show that A is open in X.
(b) If A is closed in Y, and Y is closed in X, show that A is closed in X.
• March 1st 2009, 01:44 AM
aliceinwonderland
Quote:

Originally Posted by horowitz
Let X be a topological space, and let Y subset of X have the subspace topology.
(a) If A is open in Y, and Y is open in X, show that A is open in X.
(b) If A is closed in Y, and Y is closed in X, show that A is closed in X.

(a) If A is open in Y, then $A=Y \cap U$ for an open set U in X by the definition of a subspace topology. An intersection of open sets is an open. Thus, A is open in X.

(b) is similar to (a).