pi-base
Theorem Suggestion: Paracompact + pseudocompact => compact
Theorem Suggestion
If a space is:
then it is P16.
Rationale
Negative answer to the search https://topology.pi-base.org/spaces?q=+Paracompact+%2B+Pseudocompact+%2B+%7ECompact
Proof sketch
An unbounded function can be constructed using partition of unity subordinate to an open cover with no finite subcover. It has to consist of infinitely many non-zero functions (we multiply a sequence of them by increasing numbers) and for a finite family of them there is a point outside of their supports, "leading to a large value".
Cound not find better reference than https://dantopology.wordpress.com/2012/08/25/completely-regular-spaces-and-pseudocompact-spaces/
Not sure whether Hausdorff is necessary, but the proofs I know use it (compactness is commonly considered within Hausdorff context).
This particular theorem doesn't need to be added, see here. You didn't search correctly, that's why nothing came up.
However, the search for a paracompact pseudocompact space which is not compact reveals nothing, so we may want to figure out if there exist such spaces or not.
I don't understand OP's current argument, but Fully normal + pseudocompact => compact seems like it should be able to be proven the same way
Note that there was #778 where we discussed the property Feeble compact, equivalent to pseudocompactness for Tychonoff spaces.
paracompact + feebly compact iff compact
Let me note here that the $T_2$ assumption is necessary. The easiest counterexample I can think of can be described informally as follows: Take a zigzag shape stretching infinitely to left and right. Each vertex corresponds to a point, and if two points are linked by a straight line segment, we declare the lower point is smaller than the upper point. This defines a poset. Take the Alexandrov topology associated with it. Then the space is pseudocompact - every straight line segment corresponds to a two-point subspace which is hereditarily connected (it’s just the Sierpinski space, of course), so any map from this two-point subspace into the reals has to be constant. But all points are linked through these straight line segments, so any map from $X$ to the reals is constant (in fact, this even shows any map from $X$ to a $T_1$ space is constant). It is also paracompact, as the minimal open neighborhoods of all the lower points form an open cover that refines any given open cover, and this open cover is locally finite. But it is not compact as the open cover above has no finite subcover.
One can even construct a $T_1$ counterexample, though it will be slightly more elaborate. One can modify the above and amplify the maximal points into $\omega$ to avoid maximal points becoming clopen. Then take the join of the Alexandrov topology with the cofinite topology. Each straight line segment is then hyperconnected, so again any map from a straight line segment into the reals is constant. Again, all points are connected through these straight line segments, so any map from $X$ to the reals (or more generally any $T_2$ space) is constant. Paracompactness can be shown as well. Given an open cover, first choose basic open sets covering all minimal $x$ (i.e., sets which are cofinite in $\{y: y \geq x\}$). Then this open refinement only misses finitely many points on each straight line segment. Choose basic open sets containing each of those points. This creates an open refinement that is finite on each straight line segment. Since each point has a neighborhood that only intersects finitely many straight line segments, this refinement is locally finite. Finally, it is not compact because the open cover consisting of all minimal Alexandrov-neighborhood of minimal points has no finite subcover.
I’ll write these up in more details in an MSE post and write a PR to add these examples at a later time. For now, I just want to report here that there’s really no weakening of the $T_2$ assumption.
FYI, the zigzag poset you describe is the Khalimsky line. Mentioned among other places in the Honari-Bahrampour reference in https://topology.pi-base.org/properties/P000205/references. Would be good to have it in pi-base.
FYI, the zigzag poset you describe is the Khalimsky line. Mentioned among other places in the Honari-Bahrampour reference in https://topology.pi-base.org/properties/P000205/references. Would be good to have it in pi-base.
@prabau Oh, nice to know this has been studied in the literature before. Are you aware of a good reference to cite describing this space? Honari-Bahrampour seems to be only using this as a already well-known example, so it doesn’t seem like a good idea to cite it for the space’s description.
I have not looked at it, but it's probably in the paper by Khalimsky et al. that is referenced in Honari-Bahrampour.
@david20000813 I think it's originally from either a 1969 or a 1970 Russian language paper by Khalimsky (I found the 1969 paper, but the 1970 one eludes me). The paper referenced by Honari-Bahrampour which was co-authored by Khalimsky actually doesn't seem to define it. Honari-Bahrampour seems to have the clearest direct statement of what it is, at least among the easily accessible references.
I looked through the references on Google scholar which appear under "khalimsky line". There are some nice things, but didn't find anything too optimized for our purposes. "Digital line" is a synonym for "Khalimsky line", so any reference about digital topology could be good as well, but I also didn't find it clearly and directly defined in a source on digital topology. I think Honari-Bahrampour is the best bet for a simple reference to a definition.
@GeoffreySangston Great many thanks for searching through the literature for this! I suppose I'll have to reference Honari-Bahrampour then.
(I was traveling and spending time with friends, so I just saw your comment. I probably won't have a large enough chunk of time to write a PR for this until at least after Christmas, if not after New Year, but once I get time I'll get to work adding the Khalimsky line as well as the modified version for the $T_1$ counterexample.)