Pseudocompactness-like property

[Moniker1998]

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?

[prabau]

That would be a question for Math stack exchange, or even better, for mathoverflow.

[Moniker1998]

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

1. compact implies $P_2$
2. $P_2$ implies $P_1$
3. $P_1$ implies pseudocompact
4. pseudocompact + completely regular implies $P_2$
5. paracompact + $P_1$ iff compact

[StevenClontz]

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)

[Moniker1998]

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

[Moniker1998]

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

[Moniker1998]

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?

[prabau]

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.