Editing Compact space (section)

== Basic examples ==

Any [[finite topological space|finite space]] is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed) [[unit interval]] {{closed-closed|0,1}} of [[real number]]s. If one chooses an infinite number of distinct points in the unit interval, then there must be some [[accumulation point]] among these points in that interval. For instance, the odd-numbered terms of the sequence {{nowrap|1, {{sfrac|1|2}}, {{sfrac|1|3}}, {{sfrac|3|4}}, {{sfrac|1|5}}, {{sfrac|5|6}}, {{sfrac|1|7}}, {{sfrac|7|8}}, ...}} get arbitrarily close to&nbsp;0, while the even-numbered ones get arbitrarily close to&nbsp;1. The given example sequence shows the importance of including the [[boundary (topology)|boundary]] points of the interval, since the [[Limit of a sequence|limit points]] must be in the space itself — an open (or half-open) interval of the real numbers is not compact.  It is also crucial that the interval be [[bounded set|bounded]], since in the interval {{closed-open|0,∞}}, one could choose the sequence of points {{nowrap|0, 1, 2, 3, ...}}, of which no sub-sequence ultimately gets arbitrarily close to any given real number.

In two dimensions, closed [[Disk (mathematics)|disks]] are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, an open disk is not compact, because a sequence of points can tend to the boundary – without getting arbitrarily close to any point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point ''within'' the space. Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.