The ordinary-language word ‘compact’ is used in mathematics to mean ‘lacking intermediacy’. A lack of intermediacy is sometimes obvious, such as the gaps between the integers, and sometimes surprising, as in the case of IMO 1988 problem 6: while some positive integers are squares, and some are not, the positive integer quotient referenced in IMO 1988 problem 6 is always a square. Therefore, IMO 1988 problem 6 can be considered a compactness result.
Here is a link to the statement and solution of IMO 1988 problem 6.
keywords: International Mathematics Olympiad (IMO)