Question about quotient space of K-Topology

Question

Let $R_K$ denote real line with K topology i.e. with basis consisting of all $(a,b)$ and all $(a,b)-K$ where K ={1/n, n is a positive integer}.

Look at A = $R_K/K$ i.e. identifying the whole of K to a point. Let B = ($R_K$ x $R_K$)/(K x K) i.e. quotienting out the subspace KxK.

Is B homeomorphic to AxA?

I know that there is a bijective continuous map from B to A x A by property of quotient map. I also know that the product of the quotient map q from $R_K$ to $R_K$/K with itself is not a quotient map.

How do I answer the homeomorphism question?

Answer 1

They are not homeomorphic. Let $p$ be the point of $B$ corresponding to $K\times K$, and let $q$ be the point of $A$ corresponding to $K$.

• Show that $B$ has exactly two points at which the topology is not first countable, the origin and $p$.
• Show that $A\times A$ has infinitely many points at which the topology is not first countable, namely, all of the points $\langle x,y\rangle$ such that at least one of $x$ and $y$ is in $\{0,q\}$.