Exercise · ZW-SUBSPACE-TOPOLOGY-006 · ★★☆ Intermediate
With the usual \(\mathbb{R}\) and \(A=[0,1]\), which of these sets is open in the subspace \(A\)?
Choose one option.
Show answer
- D. \([0,\tfrac12)\)
Explanation
\([0,\tfrac12)=(-1,\tfrac12)\cap A\) with \((-1,\tfrac12)\) open in \(\mathbb{R}\). By contrast, no open interval of \(\mathbb{R}\) containing \(\tfrac12\) or \(0\) meets \(A\) only in \(\{0\}\), \([\tfrac12,1)\) or \([\tfrac12,1]\).
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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