I am wondering if there is any study of "first order topological properties" that are preserved by continuous maps (between general topological spaces, or spaces satisfying mild conditions like Hausdorffness or whatever, or "nice" spaces).
I don't really have a formalization of "first order topological properties", but probably something along the lines of: in the context of a topological space $(X,\mathscr T)$, properties that $(X,\mathscr T)$ can satisfy or not satisfy, described only using notations like $\in$ (element of), $\subseteq$ (subset of),$\smallsetminus$ (set minus), etc. and also set theoretic things like $|\bullet |$ (cardinality), $\bigcup$ (union), $\times$ (Cartesian product), functions, and references to "standard" sets/spaces like $\mathbb N$ or $[0,1]$ (and of course all the standard first order logic things like $\forall$ (for all), $\exists$ (there exists), $=$ (equals), etc.).
Most famously (Topological properties preserved by continuous maps -> https://en.wikipedia.org/wiki/Continuous_function#Properties_2), we have
- $\forall \mathscr U \subseteq \mathscr T$ s.t. $X \subseteq \bigcup \scr U$, $\quad\exists \mathscr U' \subseteq \mathscr U,\exists n\in \mathbb N$ with $|\mathscr U'|=n$ and $X \subseteq \bigcup \mathscr U'$.
- $\exists U \in \mathscr T$ s.t. $(X\smallsetminus U) \in \mathscr T$.
- $\forall x,y\in X, \exists$ continuous $\gamma:[0,1]\to X$ s.t. $\gamma(0)=x$ and $\gamma(1)=y$.
- $\exists d:\mathbb N \to X$ s.t. $\forall U \in \mathscr T$, $\quad \exists n\in \mathbb N$ s.t. $d(n) \in U$.
otherwise known as compactness, connectedness, path-connectedness, separability. Of course we can have minor variations of these --- e.g. for compactness we also have
- Lindelof property: $\forall \mathscr U \subseteq \mathscr T$ s.t. $X \subseteq \bigcup \scr U$, $\quad\exists \mathscr U' \subseteq \mathscr U,\exists$ a bijection $b:\mathbb N\to \mathscr U'$ and $X \subseteq \bigcup \mathscr U'$
- countable compactness: $\forall \mathscr U \subseteq \mathscr T$ s.t. [$X \subseteq \bigcup \scr U$ and $\exists$ a bijection $b: \mathbb N \to \mathscr U$], $\quad\exists \mathscr U' \subseteq \mathscr U,\exists n\in \mathbb N$ with $|\mathscr U'|=n$ and $X \subseteq \bigcup \mathscr U$
- $\sigma$-compactness, sequential compactness, etc.
These seem so "hodge-podge"; there surely must be other simple properties we can formulate (that may not have such nice intuitive meanings as the above notions)! And surely someone has made a systematic study of such notions!
Hopefully there are some as-yet undiscovered such properties that are "useful" in the sense that "nice" sets in "nice" spaces fall in both camps: satisfy and does-not-satisfy (e.g. compactness and connectedness are "useful" because we come across some sets that are compact, and some sets that are not compact all the time when working in say $\mathbb R^d$ and these properties allow us to distinguish them topologically).
