Page 75 - DMTH503_TOPOLOGY

Unit 7: Continuous Functions

Definition: Continuity on a set. A function Notes
f :(X,T) (Y,U)

is said to be continuous on a set A  X if it is continuous at each point of A.

Notes The following have the same meaning:
(a) f is a continuous map.
(b) f is a continuous relative to T and U
(c) f is T  U continuous map.

Example 1: Let  denote the set of real numbers in its usual topology, and let  denote

the same set in the lower limit topology. Let
f :   

be the identity function;
f(x) = x for every real number x.
then f is not a continuous function; the inverse image of the open set [a, b) of  equals itself,

which is not open in . On the other hand, the identity function.
g :   

is continuous, because the inverse image of (a, b) is itself, which is open in  .


7.1.2 Homeomorphism

Definition: A map f : (X, T)  (Y, U) is said to be homeomorphism or topological mapping if
(a) f is one-one onto.

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(b) f and f are continuous.
In this case, the spaces X and Y are said to be homeomorphic or topological equivalent to one
another and Y is called the homeomorphic image of X.

Example 2: Let T denote the usual topology on R and a any non-zero real number. Then
each of the following maps is a homeomorphism
(a) f : (R, T)  (R, T) s.t. f(x) = a + x

(b) f : (R, T)  (R, T) s.t. f(x) = ax
(c) f : (R, T)  (R, T) s.t. f(x) = x where x  R.
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Example 3: Show that (R, U) and (R, D) are not homeomorphic.
Solution: Every singleton is D-open and image of a singleton is again singleton which is not
U-open. Consequently no one-one D  U continuous map of R onto R can be homeomorphism.
From this the required result follows.

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