Continuity and constructions · Definition
Quotient topology
Before you start you need to know
It is the way to glue or identify points of a space while still having a topological space.
Definition
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 \(X\).
When points of a space are identified, for example the two ends of a segment, one gets a surjective function \(q:X\to Y\) sending each point to its class. The quotient topology declares a set of classes open when the union of the corresponding classes is open in \(X\), that is, when \(q^{-1}(V)\) is.
With this definition \(q\) is continuous, and the quotient topology is the finest one that achieves this. The key property is universal: a function \(f:Y\to Z\) is continuous exactly when \(f\circ q\) is continuous. So, to define a continuous function out of a glued space, it suffices to define it on \(X\) so that it is constant on each class.
A guiding example: \([0,1]\) with \(0\sim 1\) gives a circle, and a square with opposite sides glued gives a torus. Beware: a quotient can lose properties; for instance, a quotient of a Hausdorff space need not be Hausdorff. The typical mistake is to require \(q(V)\) to be open instead of \(q^{-1}(V)\).
It builds on continuous functions and prepares the construction of surfaces and spaces by gluing, as well as the study of homogeneous spaces.
Think about it
If \(X\) is discrete, what is any quotient of \(X\) like?
Discrete: every preimage is open, so every subset of the quotient is open.
Why is \(q\) continuous for the quotient topology?
Because the open sets of the quotient were defined precisely as the \(V\) with \(q^{-1}(V)\) open.
Exercises 10
- With \(X=\{1,2,3,4\}\) with open sets \(\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\), \(Y=\{a,b,c\}\) and \(q(1)=q(2)=a\), \(q(3)=b\), \(q(4)=c\), how many open sets does the quotient topology on \(Y\) have?
- With \(X=\{1,2,3,4\}\) with open sets \(\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\), \(Y=\{a,b,c\}\) and \(q(1)=q(2)=a\), \(q(3)=b\), \(q(4)=c\), which of these subsets of \(Y\) is open in the quotient topology?
- Let \(q:X\to Y\) be surjective and \(X\) a topological space. When is a set \(V\subseteq Y\) open in the quotient topology?
- Identifying the endpoints of the interval \([0,1]\) (that is, \(0\sim 1\)) with the quotient topology gives a space homeomorphic to…
- Among all topologies on \(Y\) that make \(q:X\to Y\) continuous, the quotient topology is…
- With \(X=\{1,2,3,4\}\) with open sets \(\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\), \(Y=\{a,b,c\}\) and \(q(1)=q(2)=a\), \(q(3)=b\), \(q(4)=c\), which of these subsets of \(Y\) is closed in the quotient topology?
- Let \(X=\{1,2,3,4\}\) have the discrete topology and \(q:X\to\{a,b\}\) be given by \(q(1)=q(2)=a\), \(q(3)=q(4)=b\). How many open sets does the quotient topology have?
- Let \(X=\{1,2,3\}\) have open sets \(\varnothing,\{1\},\{1,2\},X\) and \(q:X\to\{a,b\}\) with \(q(1)=a\), \(q(2)=q(3)=b\). How many open sets does the quotient topology have?
- Let \(X=\{1,2,3\}\) have open sets \(\varnothing,\{1\},\{2\},\{1,2\},X\) and \(q:X\to\{a,b\}\) with \(q(1)=q(2)=a\), \(q(3)=b\). What are the open sets of the quotient topology?
- With \(X=\{1,2,3,4\}\) with open sets \(\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\), \(Y=\{a,b,c\}\) and \(q(1)=q(2)=a\), \(q(3)=b\), \(q(4)=c\), let \(Z=\{0,1\}\) have open sets \(\varnothing,\{1\},Z\). How many functions \(f:Y\to Z\) are continuous for the quotient topology?
References
- James R. Munkres, Topology, 2nd edition, Pearson, 2000. §22
- Stephen Willard, General Topology, Dover reprint, Dover Publications, 2004. §9