The image of first countable space

Question

I'm trying to show that the continuous first countable image of a space does not have to be first countable, but the continuous open image of first countable space is first countable.

So I let $X$ be a discrete topology so that any function defined on $X$ is continuous. Also let X be first countable space, and Y a space and let $f:X \to Y $ be a continuous function.

Let $\mathscr B_x$ be a local base (countable nbh base of $x$), and pick $y\in Y $ such that $x\in f^{-1}(y)$.

If I let $W$ be a neighborhood of $y$, then what relationship to the space $X$ I can get from here, since it's only a nbh?

If I let $V$ be an open neighborhood of $y$, then since $f$ is continuous, I can get an open neighborhood $B$ of $x$ such that $f(B)\subset V$, right?

Can I get a set $\mathscr {V}=\{f(B): B\in \mathscr{B}_x\}$ to be a neighborhood base of $y$ from here?

Which part of the question are you trying to answer there at the end? — Brian M. Scott, Apr 06 '13 at 22:41
the second part, but how do I show the first part, it's kinda obvious? — Akaichan, Apr 06 '13 at 22:57

score 3 · Accepted Answer · answered Apr 06 '13 at 23:05

3

You have a reasonable idea for the first part, but you need actually to carry it out. Specifically, you need a space that isn’t first countable to use as $Y$, the codomain of your function. A very simple example is any uncountable set with the cofinite topology. And for $X$ you can then take the discrete topology on the same set. What should you use then for your continuous function $f$?

You’re on the right track for the second part. Start with an arbitrary $y\in Y$ and any $x\in X$ such that $f(x)=y$, and let $\mathscr{B}_x$ be a countable base at $x$, exactly as you’ve done. Let $$\mathscr{V}_y=\{f[B]:B\in\mathscr{B}_x\}\;,$$ since $f$ is open, you know that $\mathscr{V}_y$ is a family of open nbhds of $y$, whether it’s a base or not. Now try to prove that it is a base at $y$. Start with an arbitrary open nbhd $U$ of $y$.

What does continuity of $f$ tell you about $f^{-1}[U]$?
Now use the fact that $\mathscr{B}_x$ is a base at $x$.

answered Apr 06 '13 at 23:05

Brian M. Scott

616,228

Should I usean uncountable set with discrete topolofy for part 1? – Akaichan Apr 06 '13 at 23:38
1

@IvordesGreenleaf: Exactly: use one uncountable set, give it the discrete topology for $X$, and give it the cofinite topology for $Y$. – Brian M. Scott Apr 06 '13 at 23:51
I ended up having that there exists $f(B)\in \mathscr V_y$ such that $y\in f(B)\subset U$, and that should do it right. – Akaichan Apr 07 '13 at 01:08
@IvordesGreenleaf: It does indeed. The crucial facts here are that $f[B]$ is open, and that if $B\subseteq f^{-1}[U]$, then $f[B]\subseteq U$. – Brian M. Scott Apr 07 '13 at 01:12
Thank you very much for your time, I'm at work and trying to do this at the same time so it takes so long :) – Akaichan Apr 07 '13 at 01:28
@IvordesGreenleaf: You’re welcome. – Brian M. Scott Apr 07 '13 at 01:53

The image of first countable space

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