finite complement topology
Let be a set. We can define the finite complement topology on by declaring a subset to be open if is finite, or if is all of or the empty set. Note that this is equivalent to defining a topology by defining the closed sets in to be all finite sets (and itself).
If is finite, the finite complement topology on is clearly the discrete topology, as the complement of any subset is finite.
If is countably infinite (or larger), the finite complement topology gives a standard example of a space that is not Hausdorff (each open set must contain all but finitely many points, so any two open sets must intersect).
In general, the finite complement topology on an infinite set satisfies strong compactness conditions (compact, -compact (http://planetmath.org/SigmaCompact), sequentially compact, etc.) since each open set in a cover contains ”almost all” of the points of . On the other hand, the finite complement topology fails all but the simplest of separation axioms since, as above, is hyperconnected under this topology.
The finite complement topology is the coarsest T1-topology on a given set.
|Title||finite complement topology|
|Date of creation||2013-03-22 14:37:54|
|Last modified on||2013-03-22 14:37:54|
|Last modified by||mathcam (2727)|