# First occurrence of “by the usual compactness argument”?

In his blog, Jeff Shallit asks, what was the first occurrence of the exact phrase, "by the usual compactness arguments," in the mathematical literature?

He reports that the earliest appearance he has found was in a paper from 1953: it's on page 400 of John W. Green, Pacific Journal of Mathematics 3 (2) 393-402.

I found another example from that same year, on page 918 of F. A. Valentine, Minimal sets of visibility, Proc Amer Math Soc 4, 917-921.

So, is this year the 60th anniversary of the first appearance of that phrase?

If there was an earlier occurrence of the equivalent phrase in a language other than English, that, too, would be of interest.

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Why do you insist on exactness of this type of a phrase? – Włodzimierz Holsztyński Sep 30 '13 at 7:27
Question probably better directed to Jeff. I suppose anything showing that compactness arguments had become so standard that authors felt they could get away without giving any details would be of interest. – Gerry Myerson Sep 30 '13 at 9:36

There is a paper, written in German, called Ueber Räume mit verschwindender erster Brouwerscher Zahl. by Urysohn and Alexandroff from 1928. (Notice that Urysohn drowned while swimming with Alexandroff in 1924)

The following quote is from page 810 (emphasis added):

Der Beweis dieser Tatsache ist wörtlich derselbe wie im Falle, wo $R$ der $R^n$ ist: es genügt zu zeigen, dass die erwähnte Trennungseigenschaft im Brouwerschen Sinne induktiv ist (was aus der Kompaktheit von $R$ in der üblichen Weise folgt), und dann den Phragmén-Brouwerschen Satz anzuwenden.

And here is my translation of that quote:

The proof of this fact is literally the same as in the case, where $R$ is $R^n$: it is sufficient to show that the mentioned separation property is inductive in the sense of Brouwer (which follows from the compactness of $R$ in the usual fashion), and then apply the Phragmén-Brouwer Theorem.

I suspect, that as soon as there was a good notion of compactness, people used it in a routine way as an standard argument.

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Lets add that it was Alexandrov and Urysohn who introduced (invented) the general modern notion of compactness (called by them bi-compactness). – Włodzimierz Holsztyński Sep 30 '13 at 18:10
Yes, I guess this was sometimes around 1923/24, in some French and German papers. – Moritz Firsching Sep 30 '13 at 20:03

As far as English, there are earlier results for the phrasing standard compactness argument(s).

In particular, a few papers are linked to from 1947: see here on Google Scholar.

The earliest of the three is:

Ambrose, W. (1947). Direct sum theorem for Haar measures. Transactions of the American Mathematical Society, 61(1), 122-127.

Specifically: Lemma 4 (p. 125) contains the phrasing "by standard compactness arguments"; see below.

Historical remark: There is a French paper entitled, "History of the Concept of Compactness" (Pier, J. P. (1980). Historique de la notion de compacité. Historia mathematica, 7(4), 425-443. http://www.sciencedirect.com/science/article/pii/0315086080900063), and among the important names in its development is Alexandroff (cited in the other response here). The reference list includes two of his papers (including another one with Urysohn) from 1924, as well as a book (with Hopf) from 1935. The paper lists Fréchet as the first to define a compact set, so I wouldn't be surprised if there are earlier uses of equivalent phrasing in French that predate either of the responses thus far.

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