# Finiteness as a motivation for compactness

Another history question, and I am not sure if I will get any answers. (If anyone knows of a good history of math list to use for this question I would be happy for any tips. The one I used to post to is now closed.)

This question deals with the motivation for compactness. I wrote a paper on this topic some time back, and one of the reviewers posed a very hard question that I am trying to answer, namely whether finiteness was a motivation for compactness.

We know Frechet coined the term compact in 1904. We also know compactness related ideas (sequential, open cover, etc) were around for many decades before this. We also know that in current thinking of compactness, there are quite strong and obvious ties between compactness and finiteness (including the joke, attributed to H. Weyl: what is a compact city? it is a city that can be guarded by finitely many near-sighted policemen!)

This topic has been discussed on MO and some examples given in a classic paper of Hewitt. But neither of these discussions gets at the question of historical origin.

Does anyone know whether (and how and at what stage and to what extent) finiteness became a motivation for compactness? Is this just an after-the-fact phenomena, or was the idea of finiteness around before compactness became formally defined?

Thanks!

• I think your first link (1) was supposed to point somewhere else. Aug 12, 2014 at 11:39
• The second link doesn't get you anywhere either without a password. Can you include a direct link to jstor instead of your university's proxy? Aug 12, 2014 at 11:45
• I have (at least tried to) fix the links now. Let me know if there is still a problem! Aug 13, 2014 at 6:33

According to Wikipedia, which first talks about the other not-finite-sounding notions,

However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. The significance of this lemma was recognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904).

• Thanks. Yes this is the idea that compactness is essentially finiteness. I think that Hadamard (Frechet's advisor) must have also had a similar idea because he suggested to Frechet to use neighborhoods to define compactness rather than nested intersections. Not exactly sure if this is enough evidence that compactness came about as way to capture finiteness in abstract spaces, or if it was more like an idea that emerged after some time, and is clear in retrospect. Thanks again for this quote! Aug 13, 2014 at 9:22

This is not strictly an answer to the history question, but I would like to take this opportunity to record my dissent from the common assertion that compactness is about finiteness.

Consider the relation $K\subset U$ as a function (predicate) of the argument $U$ ranging over open subspaces. This function is continuous in $U$ (with respect to the Scott topology) iff $K$ is compact.

If you haven't heard of the Scott topology before, you can take this equivalence as its definition (assuming that you know the usual one for compactness!).

When we also consider membership $x\in U$ as a predicate $\phi(x)$, the containment $K\subset U$ becomes $\forall x:K.\phi(x)$.

So a compact subspace is one over which we may perform universal quantification.

The dual treatment of the existential quantifier is described, using the novel notion of overtness in this answer by @Andrej Bauer, and the additional one by me tries to relate the idea to familiar ones such as the Newton-Raphson algorithm.

In print, compactness and overtness in $\mathbb R$ are explored in the paper by Andrej and me and the further one of mine.

• Thanks for the comment. I will be interested to see what others think about it. To me, it is not immediately clear that covering and finiteness are at odds. But you, I am sure, know more about this than I do! Feb 16, 2015 at 14:19
• Surely there is no conflict between your statement "the true mathematical heart of the idea of compactness is 'covering'" and the popular slogan "the historical motivations for the idea of compactness were 'finiteness'"? May 13, 2016 at 2:51