Continuous images of compact spaces

Question

This question is related to An example of a compact topological space which is not the continuous image of a compact Hausdorff space?.

Notation: A quasicompact space is one such that each open cover has a finite subcover. A compact space is a Hausdorff quasicompact space. Any space $Y$ occurring as the range of a continuous surjection $f : X \to Y$ defined on a compact $X$ is said to be a continuous image of a compact space or abbreviated a cic-space (this is an ad-hoc-notation, if there should be an "offical" term I would be glad to learn this).

Facts:

(1) cic-spaces are quasicompact.

(2) Not every quasicompact space is a cic-space (see the question mentioned above).

This leaves the question what is known about the class of cic-spaces. In particular, is there an internal characterization by suitable axioms?

It seems that separation axioms will not play a role because any space with the trivial topology is a cic-space.

I was not able to find something substantial in the literature and my own efforts did not produce a tangible result.

Answer 1

Let $\mathscr{C}$ be the minimal class of all topological spaces such that

• If $X$ is compact Hausdorff, then $X \in \mathscr{C}$.
• If $X \in \mathscr{C}$ and $f: X \to Y$ is a surjective continuous function, then $Y \in \mathscr{C}$.

(This is a common way to define classes of spaces in what's called "categorical topology".) It's clear that all cic spaces are in $\mathscr{C}$, and the reverse also holds, because the class $\mathtt{CIC}$ of all cic spaces is a class that obeys the two properties (a continuous image of a cic space is a cic space), so by minimality, $\mathscr{C} \subseteq \mathtt{CIC}$.

Other obvious properties:

• $\mathscr{C}$ is closed under arbitary products (as compact Hausdorff spaces are, and product maps are continuous.) In general, I think that any limit construction that $\mathtt{CompHaus}$ (the full subcategory of $\mathtt{Top}$ of compact Hausdorff spaces) is closed under, $\mathscr{C}$ will also be closed under, e.g. like finite sums.
• $\mathscr{C}$ is closed under closed subspaces: if $f: Y \to X$ is a continuous surjective map and $X$ is in $\mathscr{C}$ and $A \subseteq X$ is closed, use the restriction $f: f^{-1}[A] \to A$ of the compact Hausdorff $f^{-1}[A]$ to see $A \in \mathscr{C}$.

We saw in the linked post that $\mathtt{KC} \cap \mathscr{C} = \mathtt{CompHaus}$, i.e. a cic space that is KC is Hausdorff.

Proof: let $X$ be the continuous image of a compact (and Hausdorff) space $C$ under a function $f: C \to X$. Suppose $X$ is KC and $x_1 \neq x_2$ are two points of $X$. Then, as KC spaces are $T_1$ and $f$ is continuous, $F_1 = f^{-1}[\{x_1\}]$ and $F_2 = f^{-1}[\{x_2\}]$ are disjoint closed subsets of $C$, which is (as is well-known) a normal ($T_4$) space so there are disjoint open sets $U$ and $V$ of $C$ such that $F_1 \subseteq U$ and $F_2 \subseteq V$. Then $O_1 = X\setminus f[C\setminus U]$ is open in $X$ as $X$ is KC, and so is $O_2 = X\setminus f[C\setminus V]$. Also, $O_1$ and $O_2$ are disjoint open neighbourhoods of $x_1$ and $x_2$. So $X$ is Hausdorff.

I don't know yet whether any injective map between spaces in $\mathscr{C}$ is a homeomorphism (as is the case in $\mathtt{CompHaus}$). [update] As Paul Frost noted, it's easy to see that counterexamples exist: all finite spaces are in $\mathscr{C}$ being the continuous image of a finite discrete space, and amobg finite spaces there are obvious continuous bijections that are not homeomorphisms. It is clear that continuous bijections are homeomorphisms holds when the codomain $Y$ has the (decidedly articifical) property that every cic subspace of $Y$ is closed (a weakening of KC), because then such a bijection becomes a closed map (and injective closed and continuous maps are homeomorphisms).

These simple observations don't yet give an internal characterisation of this class of spaces, though, but maybe they inspire people to come up with one. It semetimes does happen that such an "external" property has an internal characterisation, as the case of $H$-closed spaces shows: this is defined as the class of Hausdorff spaces $X$ such that if $i: X \to H$ is an embedding of $X$ into any Hausdorff space $H$, then $i[X]$ is closed in $H$. This is an "external" definition, as we need consider the relation of $X$ to other Hausdorff spaces to define it, but it's also equivalent to "every open cover of $X$ has a finite subfamily with dense union", a nicely "internal" definition.