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Unit 7: Continuous Functions

 A map f : (X, T)  (Y, U) is said to be homeomorphism or topological mapping if Notes
(a) f is one-one onto.
(b) f and f are continuous.
-1
 A map f : (X, T)  (Y, U) is called an open map if it maps open sets onto open sets i.e. if any
G  T  f(G)  U.
 A map f : (X, T)  (Y, U) is called a closed map if any T-closed set F
 f(F) is U-closed set.

7.3 Keywords

Discrete Space: Let X be any non empty set and T be the collection of all subsets of X. Then T is
called the discrete topology on the set X. The topological space (X, T) is called a discrete space.
Indiscrete Space: Let X be any non empty set and T = {X, }. Then T is called the indiscrete
topology and (X, T) is said to be an indiscrete space.
Open and Closed set: Let (X, T) be a topological space. Any set A  T is called an open set and
X  A is a closed set.

7.4 Review Questions

1. In any topological space, prove that f and g are continuous maps  gof is continuous map.
Let A, B, C be metric spaces if f : A  B is continuous and g : B  C is continuous, then
gof : A  C is continuous.
2. Show that characteristic function of A  X is continuous on X iff A is both open and closed
in X.
3. Suppose (X, T) is a discrete topological space and (Y, U) is any topological space. Then
show that any map
f : (X, T)  (Y, U)
is continuous.
4. Let T be the cofinite topology on R. Let U denote the usual topology on R. Show that the
identity map
f : (R, T)  (R, U)
is discontinuous, where as the identity map
g : (R, U)  (R, T)
is a continuous map.
5. Show that the map
f : (R, U)  (R, U) given by
2
f(x) = x  x  R is not open
U-denotes usual topology.

7.5 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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