Exercise · ZW-SUBSPACE-TOPOLOGY-010 · ★★★ Advanced
If \(A\subseteq B\subseteq X\), how do the topology \(A\) inherits from \(B\) and the one it inherits directly from \(X\) compare?
Choose one option.
Show answer
- C. They always coincide
Explanation
An open set of \(A\) as a subspace of \(B\) is \(A\cap(B\cap U)\) with \(U\) open in \(X\), and since \(A\subseteq B\) this is \(A\cap U\): the same as from \(X\). No hypothesis on \(B\) or \(A\) is needed.
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-008 Let A⊆ X have the subspace topology and let V be open in A. When is V guaranteed to be open in X?