compact space

Compactness is a property that seeks to generalise the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "mising endpoints", i.e. it includes all limiting values of points.

For example, the open interval \[\left( 0,1 \right)\] is not compact as it excludes the limiting values of 0 and 1, whereas the closed interval \[\left[ 0,1 \right]\] is compact. Similarly, the space of rational numbers \[\mathbb{Q}\] is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \[\mathbb{R}\] is not compact either, because it excludes the two limiting values \[+\infty\] and \[-\infty\].

A generalisation of compactness is that a topological space is sequentially compact if every infinite sequence of points in \[X\] has a convergent subsequence converging to some point of the space. In everyday spaces like \[\mathbb{R}^{1}\] and \[\mathbb{R}^{2}\], the Bolzano-Weierstrass theorem tells us a set is sequentially compact exactly when it's closed and bounded. Hence, if you have an infinite number of points in a set like \[\left[ 0,1 \right]\], you can't avoid having some of those points get arbitrarily close to a particular point in the interval. Similarly, since neither 0 nor 1 are members of the open unit interval \[\left( 0,1 \right)\], those same sets of points would not accumulate to any point of it, so the open interval is not compact.