I am struggling to understand open maps. An open maps open sets to open sets.

Given an open map between topological spaces

$f : X \rightarrow Y$

If $U \in Y$ is open, $f^{-1}(U)$ can be open or closed as an open map can map closed sets to open sets.

What I don't understand is how if

$U \in Y$ is closed, $f^{-1}(U)$ be open.

As far as I can see this can only happen if the range of $f$ is smaller than the codomain. In other words, we can extend an open set in $U \in Y$ to a closed set $V \in Y$ with elements in the codomain whose preimage is the empty set.

Is this correct? Or are there other ways to construct an open map where the preimage of a closed set is not closed?

  • 1
    $\begingroup$ The preimage of the set $[-10,10]$ with $\arctan$ is open. $\endgroup$ – copper.hat Oct 21 '18 at 22:34
  • $\begingroup$ Let $X=Y=\mathbb{R}$ and $f$ be the identity map. Trivially, $f$ is open. Let $U=\mathbb{R}$, a set which is both open and closed. Then, $f^{-1}(U)=U$. $\endgroup$ – parsiad Oct 21 '18 at 22:36
  • $\begingroup$ @copper.hat. I don't understand your example. Although $arctan([-10, 10])$ is an infinite union of closed sets, I don't see how it is an open set in the normal topology on $\mathbb(R)$ $\endgroup$ – Jeff Oct 22 '18 at 2:25
  • 1
    $\begingroup$ You were asking about the inverse image of a closed set. Look at $\arctan^{-1}([-10,10])$ which is the entire real line. $\endgroup$ – copper.hat Oct 22 '18 at 2:30
  • $\begingroup$ @copper.hat Got it. However, as the poster below stated this is a clopen set. I am more interested in finding a closed set whose preimage is open (not clopen) or neither. $\endgroup$ – Jeff Oct 22 '18 at 2:50

Remember that "closed" is not the same as "not open." In general, in a topological space $\mathcal{X}=(X,\tau)$ there are lots of sets which are neither open nor closed; more relevantly, there may be lots of sets which are both closed and open ("clopen"). The whole space and the empty set always give examples of these - so for any space $\mathcal{X}$, the identity function on $X$ is an open map and both $X$ and $\emptyset$ are open sets with closed preimage under this map - but there are also natural spaces with lots of nontrivial clopen sets, such as Cantor space (= the space of infinite sequences of $0$s and $1$s, with the topology generated by sets of the form $\{f: \sigma\prec f\}$ for $\sigma$ a finite binary string; alternately, the Cantor set with the subspace topology as a subset of $\mathbb{R}$ with the usual topology).

  • $\begingroup$ I m not so interested in the clopen case, becasue I really was trying to understand non-continuous open maps and if $f^{-1}(U)$ is clopen with $U$ closed then $f$ isn't necessarily discontinuous. Sets that are neither closed nor open, however, are more interesting for the intuition I am trying to develop. $\endgroup$ – Jeff Oct 22 '18 at 2:21

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