This is a very loose association from someone coming from mathematical analysis background, maybe interesting for some people. In analysis/topology you have th

This is a very loose association from someone coming from mathematical analysis background, maybe interesting for some people. In analysis/topology you have this concept of a compact set. Compact sets are "small" - they are close (in a sense) to finite sets, and close (in another sense) to finite-dimensional sets, balls in infinite-dimensional Banach spaces are never compact, etc. (The notion of compactness is one of the most fundamental concept in mathematical analysis.) One of the famous fixed point theorems, Schauder fixed point theorem, asserts that a continuous mapping of a non-empty, compact and convex set has a fixed point. (This is a direct generalization of the Brouwer fixed point theorem, perhaps one of the two most well-known results of this type.) Now, in 1930 Kuratowski introduced a so-called "measure of non-compactness" - a number saying "how far the given set is from being compact". (Since then, many similar notions were also examined.) There is a very cool family of fixed-point theorems of the type "Let X be some «nice» set and f:X → X some continuous mapping which transforms sets into «less non-compact» sets; then, it has a fixed point." (At least some of those theorems require the axiom of choice, btw. Also, Kuratowski had no idea about them AFAIK, his motivation to consider his measure was completely different.) Just wondering if CS people have their way of measuring non-decidability of problems... (Probably yes.)