Page 252 - DMTH503_TOPOLOGY

Topology

Notes 1
d
Also, each such set is finite and for each xX, there is an element x m  A such that x, x m   .
m
m
i
i
Then A = A X is countable as it is the union of countable sets.
m
m N

Now AX A  X
 A X since X is closed  X = X.
In order to show that (X, d) is separable, it is sufficient to show that A = X, for which it is
sufficient to show that each point of X is an adherent point of A.

æ 1 ö
So, let x be an arbitrary point of X and G be any open nhd. of x,an open sphere S x, m ø ÷ for
ç
è
some positive integer m such that,
æ 1 ö
x S x, ÷ G …(1)
ç
è m ø
1
But for each xX,x  A A such that (x, x ) <
d
m i m m i m
æ 1 ö
or x  S x, ÷ …(2)
ç
m i è m ø
Then from (1) and (2), we get
æ 1 ö
x  S x, ÷ G.
ç
m i è m ø
Thus, every open nhd. of x contains at least one point of A and therefore, x is an adherent point
of A.

This shows that every point of X is an adherent point of A.
 X A and therefore
A = X

which follows that A is countable dense subset of X and hence X is separable.
30.2 Summary

 A family  of functions on a metric space (X, d) is called equicontinuous if  x  X,   >
0,   > 0 s.t.  y  X with d(x, y) <  , we have

f(x) f(y) <  for all f.
-
 A family  of functions on a metric space (X, d) is called uniformly equicontinuous if
  > 0,   > 0, s.t. x, yX with d(x, y) <, we have

f(x) f(y) <  for all f.
-
 Ascoli’s Theorem: Let  be a closed subset of the function space C [0, 1]. Then  is compact
iff  is uniformly bounded and continuous.

246 LOVELY PROFESSIONAL UNIVERSITY