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, (πiX)(F)=F"

  • If x0F, I get (πiX)(F)={[x]xF,xx0}{[x0]}=A. That subset is closed if and only if π1(A) is closed in X+Y. But π1(A)=π1({[x]xF,xx0}{[x0]}), that is, π1({[x]xF,xx0})π1({[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 XY=.




But {y0} does not have to be closed in Y.



Where is the problem?

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