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A standard example in elementary topology (e.g. Munkres) of a space that is limit-point compact (every infinite subset of the space has a limit point) but not compact is the minimal uncountable well-ordered set $S_\Omega$ in the order topology. One can easily show that the space is not compact, because it is not closed in its closure. But it is limit point compact, using the fact that the set has the least upper bound property. (If anyone wants the details I can write them out.)

Question is: Who first introduced this example?

I have checked without result in Counterexamples in Topology, and the history books available at my university, but I do not find any specific references. Munkres does not give a reference either. My googling proved equally unsuccessful, though I might not be searching for the right terms.

Any help would be appreciated!

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  • $\begingroup$ I am sure that Ryszard Engelking has the respective historical notes and references in his well-known monograph on General Topology. $\endgroup$
    – Wlod AA
    Commented Nov 10, 2023 at 3:19

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The earliest reference I can find is Dugundji's book Topology, published in 1966, where this result appears as Ex. 2 on page 239.

Here's how I found that. First, this result appears in Kelley's book General Topology, published in 1975, which Munkres cites on page 518. Kelley includes the result that $S_\Omega$ is compact but not locally compact in E(e) at the top of page 179.

The result also appears in Counterexamples in Topology, page 69, items 6 and 7. This was published in 1970. Both sources cite papers by Dugundji and Arens, which led me to Dugundji's book. It seems likely to me that the result is first due to Dugundji, since he and Arens were working out the connections between order topologies and compactness in that decade.

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Space $\ S_\Omega,\ $ and its generalization for higher initial ordinal numbers was the very first example of this type, and it was introduced by the authors of the notion of bicompact spaces -- by P.S. Alexandrov and P.S. Urysohn in the their classic "Memoir on compact topological spaces".

The first publication of the paper was delayed by certain circumstances while it was accepted for publication in Poland by Fundamenta Mathematicae in 1923 but published years later elsewhere, in 1929, years after Urysohn's death. And the first translation into Russian appeared in 1950, in Trudakh... Soviet Academy of Sciences.

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  • $\begingroup$ First, the question was not "when was $S_\Omega$ first introduced?" The question was: when was it first observed to be locally compact but not compact? Secondly, you shouldn't go around calling other people "naive." Read up on the Mathoverflow FAQ, and focus on the "be nice" part. This is a community of people trying to help each other, not some kind of nasty competition. Lastly, you haven't provided any way to actually verify what you wrote. A link to the relevant part of this memoir would be nice. $\endgroup$
    – David White
    Commented Nov 10, 2023 at 3:54
  • $\begingroup$ Somehow (a miracle) I still have the 3rd edition of the Memoir, published by "Nauka", Moscow 1971 (in Russian of course). I could trandlat the directly relevant lines, about the example. The intro to that edition was written by P.S.Alexandrov, 1971-03-25 -- it provides the historical background. $\endgroup$
    – Wlod AA
    Commented Nov 10, 2023 at 4:18
  • $\begingroup$ With all my respect for Arens and Dungundji, everything fundamental and original about the simple space $\ S_\Omega\ $ was said before their time. Among other related old examples there is, say, "*Tykhonov desk" (Eng?) of a Tykhonov space that is not normal. $\endgroup$
    – Wlod AA
    Commented Nov 10, 2023 at 4:29

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