Exercise · ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-009 · ★★★ Advanced
In a Hausdorff space \(X\), which statement is false?
Choose one option.
Show answer
- B. Every closed set is compact
Explanation
The closed set \(\mathbb{R}\) of \(\mathbb{R}\) is not compact. The others are true: “every compact set is closed” is the theorem, and the union and intersection of compact sets follow from it.
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:
- ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-002 Which of these subsets of ℝ cannot be compact?
- ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-003 In X=\a,b,c\ with open sets ∅,\a\,X, the set \a\ is:
- 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-005 In the proof, to separate a point p∉ K from a compact K, what is done with the disjoint open sets U…
- 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:
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