= Cofinite topology
{title2=$\tau_{\mathrm{cof}}=\{\varnothing\}\cup\{X\setminus F:F\subseteq X\text{ finite}\}$}
{wiki}
= Finite complement topology
{synonym}
The <cofinite topology> on a <set> $X$ declares the <empty set> and all <cofinite> <subsets> to be <open sets>: a nonempty <open set> may exclude only a <finite set> of points. There is no assumed <metric>. On an <infinite set>, this gives a particularly small collection of <open sets> while still making every <singleton> a <closed set>. On a <finite set>, every <subset> has finite <complement>, so the <cofinite topology> is the <discrete topology>.<Point-Set Topology by Romyar Sharifi/Chapter 1: Topological spaces>[Definition 1.1.9: finite complement topology]{ref}
For an original example, use $X=\mathbb N_{>0}$ and choose
$$
U=X\setminus\{2,5,8\},\qquad V=X\setminus\{1,3,7\}.
$$
Both are <open sets>, and their <intersection> is $U\cap V=X\setminus\{1,2,3,5,7,8\}$. Thus $4,6,9,10,\ldots$ are in both. Their finite exclusions cannot separate the infinite remainder. The <finite intersection> of any finite collection of nonempty <open sets> is again a nonempty <open set>, since its <complement> is a <finite union> of <finite sets>. An arbitrary nonempty <union> is <open> because its <complement> is contained in the finite <complement> of any one nonempty member.
\Image[/cofinite-open-sets-finite-exclusions.png]
{title=Two <open sets> with finite exclusions still overlap in an infinite <cofinite topology>}
{description=Original sets on $\mathbb N_{>0}$. Only the first sixteen labels are drawn; the dots continue indefinitely and are all included after the last excluded label. Spacing is for readability and defines no <metric>. Filled circles belong to the displayed <set>; crosses are excluded.}
The <cofinite topology> is useful for separating hypotheses that familiar <metric spaces> package together. An infinite cofinite <topological space> is a <T1 space>, because its <singletons> are <closed>, but is not a <Hausdorff space>, because two nonempty <open sets> cannot be <disjoint sets>. It is a <hyperconnected space> and hence a <connected space>. Moreover, <every subset of a cofinite space is compact>, and <sequence convergence in the cofinite topology> can have many limits. These are compatible properties, rather than contradictions: the usual uniqueness and closedness results use the <Hausdorff space> hypothesis.<Point-Set Topology by Romyar Sharifi/Chapter 1: Topological spaces>[Example 1.5.10 and Proposition 1.5.11: closed points, overlapping neighbourhoods, non-repeating sequences and uniqueness in Hausdorff spaces]{ref}
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