What to call a continuous function with preimage preserving nowhere-density?

Question

Currently I am reading some basic literature on descriptive set theory and boolean algebras. And this comes out a lot, for example in results like:

Let $X$ and $Y$ be topological spaces, and $f:X \to Y$ be a continuous function such that for every nowhere dense subset $N \subset Y$, the subset $f^{-1}(N)$ is nowhere dense. Then, there is an order-continuous boolean homomorphism between algebras or regular open sets $\pi : \mathbf{RO}(Y) \to \mathbf{RO}(X)$, defined by $\pi(U) = \operatorname{int}\overline{f^{-1}(U)}$.

I wish to call functions like $f$ in the statement by some simple short name. I'd like to call such functions meager or thick maps. Another possible name I thought about is just nowhere dense maps. Basically, I could just use something like regular map or N-map. But those are probably not good. There is some parallel with maps with preimages preserving $\sigma$-ideals of measure zero-sets. So, if there is a conventional notion for them, It would be better to use it, as meager sets can be thought as constituting the zero $\sigma$-ideal for a $\sigma$-algebra of Baire-property sets.

But now I want to ask more knowledgeable people if there is some standard terminology in some books or papers I am not aware of? I understand that there is probably no standard notions fore the whole field, but I believe that somebody already faced this problem before me. So, I am not asking to choose a name, but to share some (possibly indirect) links to existing texts (or even talks).

Sorry if this question is just unanswerable because there are no cases of such terminology.

Answer 1

The property that you are referring to has been called various names such as Skeletal, demi-open, and weakly open [1]. Furthermore, this property has quite a few characterizations, and it generalizes well to point-free topology.

Recall that a frame is a complete lattice that satisfies the identity $$x\wedge\bigvee_{i\in I}y_{i}=\bigvee_{i\in I}(x\wedge y_{i}).$$

If $L$ is a frame, then define $x^{*}=\bigvee\{y\in L\mid x\wedge y=0\}$. If $L$ is a frame, then let $\mathfrak{B}L=\{x^{**}\mid x\in L\}$. Then $\mathfrak{B}L$ is the smallest dense sublocale of $L$. Define a mapping $\beta_{L}:L\rightarrow\mathfrak{B}L$ by letting $\beta_{L}(x)=x^{**}$. Then $\beta_{L}$ is a frame homomorphism.

Theorem: Suppose that $\varphi:L\rightarrow M$ is a frame homomorphism. Then the following are equivalent:

1. There is some frame homomorphism $\psi:\mathfrak{B}L\rightarrow\mathfrak{B}M$ such that $\beta_{M}\varphi=\psi\beta_{L}$.

2. $\varphi(a^{**})\leq\varphi(a)^{**}$ for each $a\in L$.

3. $\varphi(a^{**})^{**}=\varphi(a)^{**}$ for each $a\in L$.

4. $\varphi(a^{*})^{*}\leq\phi(a)^{**}$ for each $a\in L$.

5. $\varphi(a^{*})^{*}=\phi(a)^{**}$ for each $a\in L$.

6. For each dense $a\in L$, the element $\phi(a)$ is dense in $M$.

Furthermore, if $\varphi=\Omega f$ for some $f:X\rightarrow Y$, then the above conditions are equivalent to the following conditions:

7. For each non-empty open set $U\subseteq X$, the set $(\overline{f[U]})^{\circ}$ is also non-empty.

8. For each open $U\subseteq X$, there is an open $V\subseteq Y$ such that $\overline{V}=\overline{f[U]}$.

9. If $N$ is nowhere dense in $Y$, then $f^{-1}[N]$ is nowhere dense in $X$.

10. If $U$ is a dense open subset of $Y$, then $f^{-1}[U]$ is a dense open subset of $X$.

1 http://www.numdam.org/article/CTGDC_1996__37_1_41_0.pdf B. BANASCHEWSKI. A. PULTR. Booleanization Cahiers de topologie et géométrie différentielle catégoriques, tome 37, no 1 (1996), p. 41-60

Answer 2

In the recent paper Juhász, Soukup, and Szentmiklóssy - Spaces of small cellularity have nowhere constant continuous images of small weight, a continuous function with your required property is called pseudo-open. It makes sense, I think, because the property is equivalent to the fact that $f^{-1}(A)$ is an open, dense set whenever $A$ is open and dense.

Notice, however, that in the paper both the domain and the codomain are assumed to be Hausdorff and without isolated points.