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SetTheory

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6. Recall that R_K denotes the real line in the K-topology. Let Y be the quotient space obtained from R_K by collapsing the set K to a point; let p : R_K → Y be the quotient map.

(a) Show that Y satisfies the T_1 axiom, but is not Hausdorff.

(b)Show that p x p: R_K x R_K → Y x Y is not a quotient map. [Hint: the diagonal is not closed in Y x Y, but its inverse image is closed in R_K x R_K]

And R_K is the topology with basis elements (a,b) and (a,b) - K, where K is the sequence 1/n|n is a natural number.

For (a) I am trying to prove that since the quotient space consists of a bunch of one-point sets and they are all closed, Y is T1. What does Y look like? I thought it was R_K where rather than subtracting the points in K, it was subtracting the collapsed single point set and then THIS set was being partitioned into the quotient space. Is this thinking correct, or does the subtraction of K come after R_K has been partitioned?

Assuming Y does look like this, then why must all these one point sets be closed? I remember that K is closed in R_K, but I do not see how this would imply all the one point sets to be closed. I think knowing how Y looks would help a lot.

For (b)...well I have an idea of how to approach this part.