Let X and Y be two disjoint topological spaces, x0∈X, y0∈Y and we consider the Wedge Sum (the quotient of the union by the relation x0∼y0).
I want to proof that π∘iX and π∘iY are embedding, with π:(X+Y)→(X∨Y) the canonical projection.
I have already proved that they are injective and continuous. The problem is to prove that it is closed (or open or (π∘iX)−1:(π∘iX)(X)→X continuous).
-Closed: I take F closed in X.
- If x0∉F, no problem, (π∘iX)(F)=‘‘F"
If x0∈F, I get (π∘iX)(F)={[x]∣x∈F,x≠x0}∪{[x0]}=A. That subset is closed if and only if π−1(A) is closed in X+Y. But π−1(A)=π−1({[x]∣x∈F,x≠x0}∪{[x0]}), that is, π−1({[x]∣x∈F,x≠x0})∪π−1({[x0]})=F∪{y0}.
F∪{y0} is closed if and only if (F∪{y0})∩X and (F∪{y0})∩Y are closed. (F∪{y0})∩X is closed by hypothesis, but (F∪{y0})∩Y={y0} because X∩Y=∅.
But {y0} does not have to be closed in Y.
Where is the problem?
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