pi-base
Pseudocompactness-like property
Lets call (P) to be: every locally finite open cover has a finite subcover. Its clear that (P) implies pseudocompactness, and for completely regular spaces the converse holds.
Does (P) have a common name in literature, and can we add this property to pi-base?
That would be a question for Math stack exchange, or even better, for mathoverflow.
https://mathoverflow.net/questions/479319/pseudocompact-spaces-and-locally-finite-open-covers
alright. I've distinguished 2 properties from this actually, $P_1$ and $P_2$.
$(P_1)$: Every locally finite cover has finite subcover $(P_2)$: Every locally finite cover is finite
theorems I would like to add are as follows
- compact implies $P_2$
- $P_2$ implies $P_1$
- $P_1$ implies pseudocompact
- pseudocompact + completely regular implies $P_2$
- paracompact + $P_1$ iff compact
N.B. properties that aren't explored in the literature somewhere aren't a great fit for pi-Base, unless there's strong motivation that the addition would improve the database overall (e.g. https://topology.pi-base.org/properties/P000144 helped extend results for non-T_0 spaces)
In the literature this property is seen (but not named) in Normal topological spaces by Alo and Shapiro, and also in On pseudo-compact and countably compact spaces by Iseki and Kasahara.
As for motivation, it would add that this query is impossible: https://topology.pi-base.org/spaces?q=paracompact+%2B+pseudocompact+%2B+regular+%2B+not+compact
I just got a response from K.P.Hart, the property $(P_2)$ is called feebly compact, or lightly compact according to the article Maximal feebly compact spaces by Porter, Stephenson Jr and Woods
The paper Herediarily compact spaces by Stone, or Encyclopedia of general topology list those properties, and more. But none of the ones other than maybe $P_2$ seem to have a name (i.e. feebly compact). So perhaps it'd be for the best to add "feebly compact" to pi-base, along with some theorems and examples, and maybe in some distant future to add all the different refinements of pseudocompactness, perhaps as "Stone's property X" if anything.
@prabau @StevenClontz what do you think?
Yeah, I think it would be fine to have "feebly compact" in pi-base. It's a notion that has been studied in the literature and has connections with various other properties. I'd say no need to add the other ones for now.