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!