Exercise · ZW-SUBSPACE-TOPOLOGY-005 · ★☆☆ Basic
If \(X\) has the discrete topology and \(A\subseteq X\), what topology does \(A\) have as a subspace?
Choose one option.
Show answer
- C. The discrete topology
Explanation
Every subset \(B\subseteq A\) is open in \(X\) and \(B=B\cap A\), so every subset of \(A\) is open in \(A\): it is the discrete topology.
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-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?
- ZW-SUBSPACE-TOPOLOGY-009 With the X from the first exercise, how many subsets A⊆ X with exactly 2 elements have the discrete…