Adjunction Spaces

Let X and Y be disjoint spaces. Let X union Y be the "union space" where a
set is open if and only if its intersections with X and Y are open in X and
Y resp.

Let A be a subset of X and f: A -> Y a continuous map.

Let ~ be the smallest equivalence relation on X union Y s.t. a~f(a) for all
a in A.
(the classes are {a}union f^-1(a) for a in f(A), and the single points not
in any of these sets)

This gives us an obvious quotient space X union(f) Y, and quotient map q: X
union Y -> X union(f) Y.

(I wanted to give all that to make clear what the definitions are.)

I am trying to prove that the restriction of q to Y is an embedding. The
proof of this when A is closed is straightforward. I'm not completely sure
it's true, but have been unable to find a counterexample.

Clearly q is injective and continuous. The only issue is showing that it
takes closed sets in Y to closed sets in q(Y) (or open sets). I have not
been able to show this. The main problem I'm having is that q(Y) is a
subspace of a quotient space.

Help?


.

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