Compactness reduces separating neighborhoods to finitely many

Goal and scope

Prove a compact subset K of a Hausdorff space is closed.

A general cover proof, not replaced by finite instances and not proving a converse.

Givens and symbols

• sets
• topology
• hausdorff_def
• compact_def

• G1 — X is Hausdorff and K is compact in its subspace topology.

• U_y,V_y — separating neighborhoods
• m — finite-subcover size

Rules and sources

• mko_definition · cited — For each x outside K and each y in K, choose disjoint open U_y containing x and V_y containing y. The V_y cover K, so compactness supplies a finite subcover. The corresponding finite intersection of U_y is an open neighborhood of x disjoint from K; therefore X minus K is open. Empty K is closed too. The Hausdorff hypothesis cannot be removed, and closedness does not conversely imply compactness in general. · mko-compact-subset-closed-in-hausdorff
• sets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element.
• topology · prerequisite — A topology contains empty and whole sets and is closed under arbitrary unions and finite intersections; a finite family admits exhaustive checks.
• hausdorff_def · prerequisite — Hausdorffness: every distinct pair has disjoint open neighborhoods containing the assigned points.
• compact_def · prerequisite — K is compact when every relative-open cover has a finite original-index subcover; relative opens are intersections of ambient opens with K.

Main derivation

K=\varnothing

X\setminus K=X\text{ is open}

K\ne\varnothing,\quad x\in X\setminus K

x\ne y\quad\forall y\in K

x\ne y\in K\cup\{x\}

x\in U_y,\ y\in V_y,\ U_y,V_y\text{ open},\ U_y\cap V_y=\varnothing

y\in K\cap V_y\quad(y\in K)

K\subseteq\bigcup_{y\in K}(K\cap V_y)

(K\cap V_y)_{y\in K}\text{ covers }K

K\subseteq\bigcup_{j=1}^{m}V_{y_j},\quad m\in\mathbb N

U=\bigcap_{j=1}^{m}U_{y_j}

U\text{ is open and }x\in U

z\in U\cap K

z\in V_{y_j}\cap U_{y_j}=\varnothing\text{ for some }j

\forall x\in X\setminus K\ \exists U_x\text{ open}:x\in U_x\subseteq X\setminus K

X\setminus K=\bigcup_{x\notin K}U_x\text{ open};\quad K\text{ closed}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S01, S02, S06, S07 — This establishes only the forward theorem, not that every closed set is compact.
• S07 · premise_audit · S03, S05, S06 — The index j may depend on z; no single V is assumed to cover K.
• S06 · premise_audit · S03, S05 — The finite-intersection axiom, not arbitrary intersection closure, is used.
• S05 · premise_audit · G1, S04, compact_def — Finiteness is the essential bridge to openness of the next intersection.
• S04 · premise_audit · S03 — No claim is made yet that the intersection of every U_y is open; that could be infinite.
• S03 · premise_audit · G1, S02, hausdorff_def — Neighborhoods for one y alone need not cover K.
• S02 · premise_audit · G1 — Arbitrariness of x will establish openness of the whole complement.
• S01 · premise_audit · G1 — This branch requires no point in K or nonempty subcover.

Error controls and negative controls

Finite-intersection bridge

Select finitely many V's before intersecting the corresponding U's.

Pairwise quantifiers

Separate a fixed outside point from every point of K, then release the outside-point quantifier.

• Compact subsets of every topological space are closed. → reject — A singleton in the indiscrete two-point space is compact but not closed.
• The intersection of all U_y is necessarily open. → reject — Only finite intersections are guaranteed open; compactness must be used first.
• Every closed subset of the real line is compact. → reject — R is closed; the cover (-n,n), n≥1, has no finite subcover.

Result and limitations

Under the two stated hypotheses K is closed, including empty K.

Verification and source binding

0 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 813fe0a58f2f6c26c88fc29c4ab65d01c638ff0c78dbb769f371543294dd6ab7
• original source SHA-256: 70031bac985789aa1c49e84f0a64049275096117bd97bc22051b96b5a3ce3cac
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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