Exercise · ZW-SUBSPACE-TOPOLOGY-008 · ★★☆ Intermediate
Let \(A\subseteq X\) have the subspace topology and let \(V\) be open in \(A\). When is \(V\) guaranteed to be open in \(X\)?
Choose one option.
Show answer
- B. When \(A\) is open in \(X\)
Explanation
If \(V=A\cap U\) with \(U\) open in \(X\) and \(A\) is open in \(X\), the intersection is open in \(X\). If \(A\) is not open it can fail: \([0,\tfrac12)\) is open in \([0,1]\) but not in \(\mathbb{R}\).
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
- ZW-SUBSPACE-TOPOLOGY-001 In X=\1,2,3,4\ with open sets ∅,\1\,\3\,\1,3\,\1,2,3\,X, how many open sets does the subspace topol…
- ZW-SUBSPACE-TOPOLOGY-002 With the same X and A=\2,3,4\, which of these subsets is open in the subspace A?
- ZW-SUBSPACE-TOPOLOGY-003 Let A⊆ X have the subspace topology and let i:A→ X be the inclusion. If U⊆ X, what is i^-1(U)?
- ZW-SUBSPACE-TOPOLOGY-004 With the same X, what is the subspace topology on A=\1,2\?
- ZW-SUBSPACE-TOPOLOGY-005 If X has the discrete topology and A⊆ X, what topology does A have as a subspace?
- ZW-SUBSPACE-TOPOLOGY-006 With the usual ℝ and A=[0,1], which of these sets is open in the subspace A?
- ZW-SUBSPACE-TOPOLOGY-007 With the X from the first exercise and A=\2,3,4\, which of these subsets is closed in the subspace…
- ZW-SUBSPACE-TOPOLOGY-009 With the X from the first exercise, how many subsets A⊆ X with exactly 2 elements have the discrete…