Exercise · ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-003 · ★☆☆ Basic
In \(X=\{a,b,c\}\) with open sets \(\varnothing,\{a\},X\), the set \(\{a\}\) is:
Choose one option.
Show answer
- A. compact and not closed
Explanation
It is finite, hence compact; its complement \(\{b,c\}\) is not open, so it is not closed. This does not contradict the theorem because the space is not Hausdorff.
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-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:
- ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-009 In a Hausdorff space X, which statement is false?
← ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-002 ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-004 →