4
$\begingroup$

Let $(X,\tau_x),(Y,\tau_Y)$ be topological spaces and let $f:X \to Y$ be a continuous and open function. Prove that $f(\tau_X)=\tau_Y$

This is an excercise of the problems set of my course on Introductory Topology notes. I think what I've been asked to prove is not true since if I chose $X,Y$ both equiped with the discrete topology and $Y$ having more than one element, then any constant function $f$ $(f(x)=c \in Y$ for all $x \in X)$ is continuous and open but $f(\tau_X)=\{ \emptyset, \{c\} \}\not=\tau_Y$
Is this right? Perhaps I'm missunderstanding it. Any clarification or suggestion is welcome. Thanks

$\endgroup$

2 Answers 2

4
$\begingroup$

Your counterexample is good. You could also consider the fact that $Y\notin f(\tau_X)$ unless $f$ is surjective; for instance, the inclusion of a proper open set is continuous and open.

Suppose that $f$ is surjective, besides being open and continuous. It's clear that $f(\tau_X)\subseteq \tau_Y$ (by the fact $f$ is open). If $V$ is open in $Y$, then $V=f(f^{-1}(V))$ (by surjectivity of $f$), so $V\in f(\tau_X)$ (by the fact $f$ is continuous).

$\endgroup$
3
  • $\begingroup$ Many texts assume that open maps are onto. That also ensures they are quotient maps. $\endgroup$ Feb 18, 2017 at 9:19
  • $\begingroup$ @HennoBrandsma That's why I added the proof for the case when the map is surjective. $\endgroup$
    – egreg
    Feb 18, 2017 at 9:43
  • $\begingroup$ I figured as much, yes. $\endgroup$ Feb 18, 2017 at 11:50
3
$\begingroup$

Right. In fact, a continuous function is open if and only if $f(\tau_X) \subseteq \tau_Y$, but the converse inclusion needn't hold, in general.

$\endgroup$
1
  • $\begingroup$ I see, thanks for the clarification. Actually the excercise has a second part, which asks: Is it always true that $f^{-1}(\tau_Y)=\tau_X$? I think the problem should have been stated the other way round. Probably it was intended to be : "prove $f^{-1}(\tau_Y)=\tau_X$, and is it always true that $f(\tau_X)=\tau_Y$" $\endgroup$
    – la flaca
    Feb 17, 2017 at 14:49

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .