Page 101 - DMTH503_TOPOLOGY

Unit 9: The Metric Topology

(i) Given A = A° ...(5) Notes
Aim: A is an open set
(2) and (5) A is an open set.
(ii) Given A is an open set. ...(6)
Aim: A = A°
(4) and (6) A = A°.

9.1.10 Uniform Convergence

A sequence defined on a metric space (X, d) is said to be uniformly convergent if given > 0, 
n N s.t. n n d (f (x), f (x)) <   X.
0 0 n
Theorem 7: Let <f (x) > be a sequence of continuous functions defined on a metric space (X, d). Let
n
this sequence converge uniformly to f on X. Then f(x) is continuous on X.
OR
Uniform limit of a sequence of continuous function is continuous.
Proof: Since < f (x)> converges uniformly to f on (X, d). Hence given > 0, n N independent
n 0
of x N s.t. n n .
0
 d (f (x), f(x)) < /3 ...(1)
n
Let a X be arbitrary. To prove that f is continuous on X, we have to prove that f is continuous
at x = a, for this we have to show that given > 0, > 0 s.t. d (x, a) < 
 d (f(x), f (a)) < . ...(2)
Continuity of f at a X
n

 d (f (x), f (a)) < for d (x, a) <  ...(3)
n n 3

By (1), d (f (a), f(a)) <  n  n ...(4)
n 0
3
If d (x, a) < , then
d (f(x), f(a))  d [f(x), f (x)] + d [f (x), f (a)] + d [f (a), f(a)]
n n n n
  
<     by (1), (3) and (4)
3 3 3
or d (f(x), f(a) <  for d (x, a) < . Hence the result (2).
Theorem 8: Frechet space. Let F be the set of infinite sequences of real numbers.

Let x, y, z F, then
x = <x > = <x , x ...>, y = <y >, z = <z >
n 1 2 n n
where x , y , z  R
n n n
we define a map

d : F × F :  R s.t.

 1 x  y
d (x, y) =  n n
n 1 2 n  1   x  y 

n 
n
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