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My question is the following. Suppose we have two normal topological spaces X and Y and we have a continuous map from a closed subset A of X to Y. Then we can construct another topological space by "glueing together" X and Y at A and f(A). By taking the quotient space of the disjoint union of X and Y by the equivalence relation that

x is equivalent to y if:

1) x=y

2) x,y are elements of A and f(x)=f(y)

or

3) x is an element of A and y is an element of Y and f(x)=y

or x is an element of Y and y is an element of A and x=f(y).

My question is how can you prove that this constructed space is again normal?

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# Glueing together normal topological spaces at a closed subset

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