# How can the preimage of a closed set for an open map be open?

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?

• The preimage of the set $[-10,10]$ with $\arctan$ is open. – copper.hat Oct 21 '18 at 22:34
• 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, $f^{-1}(U)=U$. – parsiad Oct 21 '18 at 22:36
• @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 $\mathbb(R)$ – Jeff Oct 22 '18 at 2:25
• You were asking about the inverse image of a closed set. Look at $\arctan^{-1}([-10,10])$ which is the entire real line. – copper.hat Oct 22 '18 at 2:30
• @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. – 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).
• 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. – Jeff Oct 22 '18 at 2:21