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There is no topology on the set of all [compact] topological spaces, because there is no set of all [compact] topological spaces. Given a set of topological spaces, consider the power set of its union …

Questions like these are easily answered with a search of pi-Base: π-Base, Search for $k_3$-space+Paracompact+~$T_2$ Six counterexamples are listed there today, including Tyrone's example. I'll sugges …

Summarizing comments as an answer. The standard interpretation of radial does not require that $\alpha\mapsto a_\alpha$ be continuous. Therefore, the Fortissimo space is radial, despite the lack of no …

Spearable implies ccc is theorem T21 of pi-Base, which references Counterexamples. The book doesn't provide a proof, but the result is standard. (Take a collection of pairwise disjoint open sets, if …

Questions like these are often answerable by a search of the pi-Base (noting that every Hausdorff paracompact space is completely uniformizable): https://topology.pi-base.org/spaces?q=%20hausdorff%2B% …

This question came to mind upon M W's answer to this recent Math.SE question - the question asked about finite spaces, but it was pointed out that all that was really needed was the Alexandrov propert …

To not leave my question unanswered, I'll note that @Tryone suggested a couple of references in comments to the question. In particular, while I do not have a copy of the paper itself, this zbMath rev …

KP answered the question, and in fact provided the answer to a stronger question: his space is strongly KC, showing that Hausdorff is quite necessary to show extremally disconnected spaces are both to …

Writing up a direct proof for Rothberger based upon Caruvana's references. It's unclear why I didn't think to try it, but it's much easier to think about the K-Rothberger game in terms of its dual - b …

Let $X$ be a set, and let $\kappa$ be an infinite cardinal. Say sets in $X$ are closed if their cardinality is at most $\kappa$. (This class includes discrete spaces as you mentioned as well as spaces …

The closed topologist's sine curve is the classic example of a connected but not locally connected space. But local neighborhoods away from the y-axis are copies of $\mathbb R$, which is connected. To …

I was visiting Auburn today and obtained a scan. https://github.com/StevenClontz/research/blob/master/miscellaneous/SKM_C650i23042612550.pdf

I don't specifically recall "almost compact" in the literature, but it's quite natural as it relates to "almost Menger" and "almost Lindelof". In general, relative compactness is not equivalent to a s …

For (1), $\omega$ bounded. I'm unaware of names for the other concepts.

The reference is here, provided you cannot find anything more classical. Every closed subset of a Menger set is Menger. Thus if $A(X)$ is Menger, then its closed subset $X\times\{0\}\cong X$ is Menger …