Topological property: set-theoretically large subsets of an infinite space are not compact.

Question

Let $X$ be an infinite topological space. Say that $X$ satisfies # if no subset of $X$ of cardinality $|X|$ is compact. So for instance it is clear that

• no (infinite) compact space satisfies #
• any infinite discrete space satisfies #
• $\mathbb R$ does not satisfy # since any nontrivial closed and bounded interval has the same cardinality as $\mathbb R$ and is compact
• any uncountable regular cardinal in the order topology satisfies #

It seems like this might be an interesting property. What do you think? Is there a name for it? Are there any other topological properties which are closely related?

Thanks

Answer 1

I haven't seen this property before (but that doesn't mean much). You could try to think how this property is preserved by standard topological constructions (but it seems it doesn't preserve well because of the dependence on size of the space). You could instead define corresponding cardinal invariant $\sup\{\lvert C\rvert: C ⊆ X \text{ compact}\}$ or its strict variant $\min\{κ: C ⊆ X \text{ is compact} \implies \lvert C \rvert < κ\}$ and see how are there invariants preserved and if there is relation to standard cardinal invariants. Note that # for $X$ is equivalent to that the strict cardinal invariant is $≤ \lvert X \rvert$.