Exercise · ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-007 · ★★☆ Intermediate
Let \(K\subseteq\mathbb{R}\) be compact and \(x_n\in K\) with \(x_n\to x\) in \(\mathbb{R}\). Is \(x\in K\)?
Choose one option.
Show answer
- A. Yes, always
Explanation
\(K\) is closed, being compact in a Hausdorff space, and a closed set contains the limits of the sequences lying in 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:
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- 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-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?
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