Monday, 11 April 2016

general topology - Wedge Sum Embedding with Inclusions

Let X and Y be two disjoint topological spaces, x0X, y0Y and we consider the Wedge Sum (the quotient of the union by the relation x0y0).



I want to proof that πiX and πiY are embedding, with π:(X+Y)(XY) 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 (πiX)1:(πiX)(X)X continuous).



-Closed: I take F closed in X.





  • If x0F, 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?

No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...