Exercise · ZW-SUBSPACE-TOPOLOGY-007 · ★★☆ Intermediate
With the \(X\) from the first exercise and \(A=\{2,3,4\}\), which of these subsets is closed in the subspace \(A\)?
Choose one option.
Show answer
- A. \(\{4\}\)
Explanation
The closed sets of \(A\) are the complements in \(A\) of its open sets \(\varnothing,\{3\},\{2,3\},A\): they are \(A,\{2,4\},\{4\},\varnothing\). Only \(\{4\}\) appears; for example, \(A\setminus\{3,4\}=\{2\}\) is not open.
Review the concept: Subspace topology
The topology \(\mathcal{T}_A\) on a subset \(A\) of a topological space \((X,\mathcal{T})\) whose open sets are the intersections of \(A\) with the open sets \(U\) of \(X\).
\[\mathcal{T}_A=\{A\cap U:\ U\in\mathcal{T}\}\]
More practice problems: subspace topology
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