Page 110 - DMTH503_TOPOLOGY

Topology

Notes Proof: For each y  Y, the set g(p ({y}) is a one-point set in Z (since g is constant on p ({y}). If we
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let f(y) denote this point, then we have defined a map f : Y  Z such that for each x  X,
f(p(x)) = g(x). If f is continuous, then g = f o p is continuous. Conversely, suppose g is continuous.
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Given an open set V of Z , g (V) is open in X. But g (V) = p (F (V)); because p is a quotient map,
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it follows that f (V) is open in Y. Hence f is continuous. If f is a quotient map, then g is the
composite of two quotient maps and is thus a quotient map. Conversely, suppose that g is a
quotient map. Since g is subjective, so is f.
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Let V be a subset of Z; we show that U is open in Z if f (V) is open in Y. Now the set p (f (V))
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is open in X because p is continuous. Since this set equals g (V), the latter is open in X. Then
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because g is a quotient map, V is open in Z.
Corollary (1): Let g : X  Z be a surjective continuous map. Let X* be the following collection of
subsets of X:
X* = {g ({z}) | z  Z}
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Give X* the quotient topology.
(a) The map g induces a bijective continuous map f : X*  Z, which is a homeomorphism if
and only if g is a quotient map.

(b) If Z is Hausdorff, so is X*.
Proof: By the preceding theorem, g induces a continuous map f : X*  Z; it is clear that f is
bijective. Suppose that f is a homeomorphism. Then both f and the projection map p : X  X* are
quotient map. So that their composite q is a quotient map. Conversely, suppose that g is a
quotient map. Then it follows from the preceding theorem that f is a quotient map. Being
bijective, f is thus a homeomorphism.

Suppose Z is Hausdorff. Given distinct points of X*, their images under f are distinct and thus
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possess disjoint neighbourhoods U and V. Then f (U) and f (V) are disjoint neighbourhoods of
the two given points of X*.
10.2 Summary

 Let X and Y be topological spaces; let p : X  Y be a surjective map. The map p is said to be
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a quotient map provided a subset U of Y is open in y if and only if p (U) is open in X.
 A map f : X  Y is said to be an open map if for each open set U of X, the set f(U) is open
in Y.

 A map f : X  Y is said to be a closed map if for each closed set A of X, the set f(A) is closed
in Y.

 If X is a space and A is a set and if p : X  A is a surjective map, then there exists exactly one
topology T on A relative to which p is a quotient map; it is called the quotient topology
induced by p.

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