Exercise · ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-005 · ★☆☆ Basic

In the proof, to separate a point \(p\notin K\) from a compact \(K\), what is done with the disjoint open sets \(U_x\ni p\), \(V_x\ni x\) given by Hausdorff?

Choose one option.

Show answer

C. Cover \(K\) by the \(V_x\) and take a finite subcover

Explanation

The \(V_x\) cover \(K\); compactness leaves finitely many \(V_{x_1},\dots,V_{x_n}\). Then \(\bigcap U_{x_i}\) is an open set containing \(p\) disjoint from \(\bigcup V_{x_i}\supseteq K\).

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Review the concept: Compact sets in Hausdorff spaces

Every compact subset \(K\) of a Hausdorff space \(X\) is closed in \(X\), and every closed subset of a compact space is compact.

\[K\subseteq X\ \text{compact},\ X\ \text{Hausdorff}\ \Rightarrow\ \overline{K}=K\]

More practice problems: compact sets in hausdorff spaces

ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-001 If X is Hausdorff and K⊆ X is compact, then K is:
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ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-004 A Hausdorff space has exactly three points. How many of its subsets are compact?
ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-006 In ℕ with the cofinite topology, which subset is compact but not closed?
ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-007 Let K⊆ℝ be compact and x_n∈ K with x_n→ x in ℝ. Is x∈ K?
ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-008 Let K_1,K_2 be compact sets in a Hausdorff space. Then K_1∩ K_2 is:
ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-009 In a Hausdorff space X, which statement is false?