show that if y is a subspace of x and A is a subset of y the

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.

Solution

Suppose that U ? A is open in the subsace topology inherited from Y .

Then there is a set V ? Y , open in the subspace topology on Y , which satisfies U = V ? A.

As V is open in the subspace topology on Y , there is an open set W ? X which satsifies V = W ? Y .

Then U(W ? Y ) ? A = W ? (Y ? A) = W ? A, and U is the intersection of an open subset of X with A.

Thus U is open in the subspace topology inherited from X.

Suppose that U ? A is open in the subspace topology inherited from X. Then there exists an open set W ? X with U = W ? A. Let V = W ? Y . Then V is an open subset of Y in the subspace topology that Y inherits from X, and V ? A = (W ? Y ) ? A = W ? A = U. So U is an open subset of A in the subspace topology inherited from Y.