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, (\pi \circ i_X)(F)=``F"
If x_0 \in F, I get (\pi\circ i_X)(F)=\{[x] \mid x \in F, x\neq x_0\}\cup\{[x_0]\}=A. That subset is closed if and only if \pi^{-1}(A) is closed in X+Y. But \pi^{-1}(A)=\pi^{-1}( \{[x] \mid x \in F, x\neq x_0\}\cup\{[x_0]\}), that is, \pi^{-1}( \{[x] \mid x \in F, x\neq x_0\})\cup \pi^{-1}(\{[x_0]\})=F\cup\{y_0\}.
F\cup\{y_0\} is closed if and only if (F\cup\{y_0\})\cap X and (F\cup\{y_0\})\cap Y are closed. (F\cup\{y_0\})\cap X is closed by hypothesis, but (F\cup\{y_0\})\cap Y=\{y_0\} because X\cap Y=\emptyset.
But \{y_0\} does not have to be closed in Y.
Where is the problem?
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