# finite complement topology

Let $X$ be a set. We can define the finite complement  topology  on $X$ by declaring a subset $U\subset X$ to be open if $X\backslash U$ is finite, or if $U$ is all of $X$ or the empty set  . Note that this is equivalent      to defining a topology by defining the closed sets  in $X$ to be all finite sets  (and $X$ itself).

If $X$ is finite, the finite complement topology on $X$ is clearly the discrete topology, as the complement of any subset is finite.

In general, the finite complement topology on an infinite set  satisfies strong compactness conditions (compact  , $\sigma$-compact (http://planetmath.org/SigmaCompact), sequentially compact, etc.) since each open set in a cover contains ”almost all” of the points of $X$. On the other hand, the finite complement topology fails all but the simplest of separation axioms  since, as above, $X$ is hyperconnected under this topology.

The finite complement topology is the coarsest T1-topology on a given set.

Title finite complement topology FiniteComplementTopology 2013-03-22 14:37:54 2013-03-22 14:37:54 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 54A05 cofinite topology  CofiniteTopology CofiniteAndCocountableTopology