Page 111 - DMTH503_TOPOLOGY

Unit 10: The Quotient Topology

 Let X be a topological space and let X* be a partition of X into disjoint subsets whose union Notes
is X. Let p : X  X* be the surjective map that carries each point of X to the element of X*
containing it. In the quotient topology induced by p, the space X* is called a quotient space
of X.

10.3 Keywords

Equivalence relation: A relation R in set A is an equivalence relation iff it is reflexive, symmetric
and transitive.
Homeomorphism: A map f : (X, T)  (Y, U) is said to be homeomorphism if (i) f is one-one onto
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(ii) f and f are continuous.
10.4 Review Questions

1. Prove that the product of two quotient maps needs not be a quotient map.
2. Let p : X  Y be a continuous map. Show that if there is a continuous map f : Y  X such that
p o f equals the identify map of Y, then p is a quotient map.
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3. Show that a subset G of Y is open in the quotient topology (relative to f : X  Y) iff f (G)
is an open subset of X.
4. Show that if f is a continuous, open mapping of the topological space X onto the topological
space Y, then the topology for Y must be the quotient topology.

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5. Show that Y, with the quotient topology, is a T -space iff f (y) is closed in X for every
1
y  Y.
6. Show that if X is a countably compact T -space, then Y is countably compact with the
1
quotient topology.
7. Show that if f is a continuous, closed mapping of X onto Y, then the topology for Y must be
the quotient topology.

8. Show that a subset F of Y is closed in the quotient topology (relative to f : X  Y) iff f (F)
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is a closed subset of X.
10.5 Further Readings

Books J.L. Kelley, General Topology, Van Nostrand, Reinhold Co., New York.

S. Willard, General Topology, Addison-Wesley, Mass. 1970.

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