# What conditions on the topological spaces are necessary and sufficient to ensure the existence of the following kind of functions?

Suppose that $$(X,\tau_X)$$ and $$(Y,\tau_Y)$$ are two topological spaces where neither is given the discrete (or indiscrete) topology.

• Does there always exists a nonconstant function $$f:X\to Y$$ is such that for all $$Z\subseteq X$$ if $$\operatorname{Cl}_Y{\left(f(Z)\right)}\subseteq f(\operatorname{Cl}_X(Z))$$ then $$f^{-1}(\operatorname{Cl}_Y{\left(f(Z)\right)})\subseteq \operatorname{Cl}_X{\left(Z\right)}$$?

• If not, then what properties are necessary and sufficient for the topological spaces to ensure the existence of such a function?

We note that if either $$f$$ is a closed continuous map or is injective then $$f$$ satisfies the condition. So if the $$X$$ and $$Y$$ has at least the same cardinality then the existence of $$f$$ is guaranteed. The remaining question thus is the following,

Suppose that $$(X,\tau_X)$$ and $$(Y,\tau_Y)$$ are two topological spaces where neither is given the discrete (or indiscrete) topology and such that the cardiality of $$Y$$ is strictly less that that of $$X$$.

• Does there always exists a nonconstant function $$f:X\to Y$$ is such that for all $$Z\subseteq X$$ if $$\operatorname{Cl}_Y{\left(f(Z)\right)}\subseteq f(\operatorname{Cl}_X(Z))$$ then $$f^{-1}(\operatorname{Cl}_Y{\left(f(Z)\right)})\subseteq \operatorname{Cl}_X{\left(Z\right)}$$?

• If not, then what properties are necessary and sufficient for the topological spaces to ensure the existence of such a function?

• Why "if $\operatorname{Cl}_Y{\left(f(Z)\right)}\subseteq \operatorname{Cl}_Y(f(Z))$" ? That's as restrictive as "if $1+1=2$" ... Apr 9, 2020 at 5:28
• @HagenvonEitzen: Nice catch. Thanks.
– user170039
Apr 9, 2020 at 5:36

If both $$X$$ and $$Y$$ are $$T_1$$ spaces then any function $$f$$ satisfying the condition is injective. Indeed, let $$x\in X$$ be any point. Put $$Z=\{x\}$$. Then $$\operatorname{Cl}_X(Z)=\{x\}$$, $$\operatorname{Cl}_Y(f(Z))=\{f(x)\}$$, thus $$\operatorname{Cl}_Y{\left(f(Z)\right)}\subseteq f(\operatorname{Cl}_X(Z))$$. Thus $$f^{-1}(f(x))=f^{-1}(\operatorname{Cl}_Y{\left(f(Z)\right)})\subseteq \operatorname{Cl}_X{\left(Z\right)}= \{x\}.$$