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Revision difference : finite complement topology |
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| Let $X$ be a set. We can define the \emph{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). |
Let $X$ be a set. We can define the \emph{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). |
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| If $X$ is finite, the finite complement topology on $X$ is clearly the discrete topology, as the complement of \emph{any} subset is finite. |
If $X$ is finite, the finite complement topology on $X$ is clearly the discrete topology, as the complement of \emph{any} subset is finite. |
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| If $X$ 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). |
If $X$ 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). |
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| In general, the finite complement topology on an infinite set satisfies strong compactness conditions (compact, \PMlinkname{$\sigma$-compact}{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. |
In general, the finite complement topology on an infinite set satisfies strong compactness conditions (compact, \PMlinkname{$\sigma$-compact}{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. |
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| The cofinite topology is the coarsest T1-topology on a given set. |
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