Characterization of discrete topology-like behavior about compact sets.

Question

It's well-known that in discrete topology a set is compact iff it is finite. There exist a lot of examples of topologies which are not discrete but with that fact still holding, and it's not hard to find some of them. Is there any theory about (infinite) topological spaces in which "compact iff finite" holds?

Answer 1

The property you refer to is called "anti-compact", and is a special case of anti-properties, of which you can find a short history here. The idea for compactness is that we know that all finite subspaces are guaranteed to be compact, and we wonder for what spaces we only have those compact subspaces which are "forced to be" compact.

So for a property $P$, we define $\operatorname{spec}(P)$ to be the class of all cardinals $\alpha$ such that

For all topological spaces $X$: if $|X| = \alpha$ then has property $P$.

So $\operatorname{spec}(\mathcal{C}) = \{\alpha: \alpha < \aleph_0\}$ when $\mathcal{C}$ is the property of being compact.

Also, $\operatorname{spec}(\text{connected}) = \{0,1\}$ e.g.

A space $X$ is then called anti-$P$ iff

For all subspaces $A \subseteq X$: $A$ has property $P$ iff $|A| \in \operatorname{spec}(P)$.

So an anti-connected space is just hereditarily disconnected. anti-compactness is just the property that any discrete space has, but e.g. also the co-countable topology. The history overview references some papers by Wilanski and Levine specifically on anti-compactness (though they don't call it that yet).