Exercise · ZW-QUOTIENT-TOPOLOGY-004 · ★☆☆ Basic
Identifying the endpoints of the interval \([0,1]\) (that is, \(0\sim 1\)) with the quotient topology gives a space homeomorphic to…
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- C. the circle
Explanation
Gluing the two endpoints of a segment closes it into a loop: a circle. A segment and a disk have no such hole, and a figure eight needs another pair of points identified, not just the endpoints.
Review the concept: Quotient topology
The topology \(\mathcal{T}_q\) on \(Y\) determined by a surjective function \(q:X\to Y\) from a space \((X,\mathcal{T})\), whose open sets are the \(V\subseteq Y\) with \(q^{-1}(V)\) open in \(X\).
\[\mathcal{T}_q=\{V\subseteq Y:\ q^{-1}(V)\in\mathcal{T}\}\]
More practice problems: quotient topology
- ZW-QUOTIENT-TOPOLOGY-001 With X=\1,2,3,4\ with open sets ∅,\1\,\2\,\1,2\,\1,2,3\,X, Y=\a,b,c\ and q(1)=q(2)=a, q(3)=b, q(4)=…
- ZW-QUOTIENT-TOPOLOGY-002 With X=\1,2,3,4\ with open sets ∅,\1\,\2\,\1,2\,\1,2,3\,X, Y=\a,b,c\ and q(1)=q(2)=a, q(3)=b, q(4)=…
- ZW-QUOTIENT-TOPOLOGY-003 Let q:X→ Y be surjective and X a topological space. When is a set V⊆ Y open in the quotient topolog…
- ZW-QUOTIENT-TOPOLOGY-005 Among all topologies on Y that make q:X→ Y continuous, the quotient topology is…
- ZW-QUOTIENT-TOPOLOGY-006 With X=\1,2,3,4\ with open sets ∅,\1\,\2\,\1,2\,\1,2,3\,X, Y=\a,b,c\ and q(1)=q(2)=a, q(3)=b, q(4)=…
- ZW-QUOTIENT-TOPOLOGY-007 Let X=\1,2,3,4\ have the discrete topology and q:X→\a,b\ be given by q(1)=q(2)=a, q(3)=q(4)=b. How…
- ZW-QUOTIENT-TOPOLOGY-008 Let X=\1,2,3\ have open sets ∅,\1\,\1,2\,X and q:X→\a,b\ with q(1)=a, q(2)=q(3)=b. How many open se…
- ZW-QUOTIENT-TOPOLOGY-009 Let X=\1,2,3\ have open sets ∅,\1\,\2\,\1,2\,X and q:X→\a,b\ with q(1)=q(2)=a, q(3)=b. What are the…