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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA [[filter|15PF73BKY6]] $\\mathcal F$ on $X$ [[converges|70M1VAPFN0]] to $x \\in X$ when every [[neighbourhood|0ZCY0Z80B6]] of $x$ belongs to $\\mathcal F$.\nThen $X$ is [[compact|0TVJZYB6SH]] if and only if every [[ultrafilter|J64Y3JC0RE]] on $X$ converges to at least one point of $X$.\n\n## Remarks\n\nSuppose $X$ is compact and an ultrafilter $\\mathcal U$ converges to no point.\nThen each $x$ has an open neighbourhood $V_x \\notin \\mathcal U$, so $X \\setminus V_x \\in \\mathcal U$.\nFinitely many $V_x$ cover $X$, and the intersection of their complements is empty but lies in $\\mathcal U$, a contradiction.\nConversely, if an open cover has no finite subcover, the complements of its members generate a proper filter; an ultrafilter that contains it cannot converge to any point.\nThis direction uses the ultrafilter lemma, a weak form of the axiom of choice.\n\nThe theorem gives a short proof of [[Tychonoff's theorem|0JTRSJKCFB]]: an ultrafilter on a product converges as soon as all its images under the projections converge.\nIf $X$ is also [[Hausdorff|0S1FJ8VEX9]], each ultrafilter has exactly one limit.\nFor example, on $X = [0, 1]$, the ultrafilter of all sets that contain a fixed point $p$ converges to $p$.\n\n## Explanations\n\n### L1 human\n\nAn ultrafilter on a space is a way of choosing, for every subset, whether it is large or small, in a consistent way.\nThe theorem says that a space is compact exactly when every such choice points to some place in the space.\nIt is the filter version of the fact that, in a compact metric space, every sequence has a convergent subsequence.\n\n### L2 human\n\nFor general spaces, sequences are not enough to describe compactness, but ultrafilters are.\nThe criterion turns proofs about open covers into proofs about limits.\nIts best-known use is the proof of Tychonoff's theorem, where convergence in a product is checked coordinate by coordinate.\nIt also leads to the Stone–Čech compactification of a discrete space as the space of ultrafilters on it.\n",
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