Page 103 - DMTH503_TOPOLOGY

Unit 9: The Metric Topology

Let x X – F be arbitrary. Then X – F is an open set containing x s.t. (X – F) F = . Notes
x is not a limit point of F
 x D (F) x X – D (F)
Thus,  x X – F  x X – D (F)

 X – F  X – D (F)
or, D (F)  F
Step II: Given D (F)  F. ...(1)
To prove F is closed.

Let y X – F, then y  F
y F, D (F)  F  y D (F)
 open set G with y G s.t.
(G – {y}) F = 

 G F = as y F
 G X – F
Thus we have show that
any y X – F  open set G with y G s.t. G X – F

 X – F is open F is closed.

9.2 Summary

 Let X be any given space. Let x, y, z, X be arbitrary. A function d : X × X R having
the properties listed below:

(i) d (x, y) 0
(ii) d (x, y) = 0 iff x = y
(iii) d (x, y) = d (y, x)
(iv) d (x, y) + d (y, z) d (x, z)
is called a distance function or a metric for X.

 Let X be any given space. Let x, y, z X be arbitrary. A function d : X × X R having
the properties listed below:
(i) d (x, y) 0

(ii) d (x, y) = 0 if x = y
(iii) d (x, y) = d (y, x)
(iv) d (x, y) + d (y, z) d (x, z), where x, y, z X is called pseudo metric on X.

+
 Let (X, ) be a metric space. Let x X and r R . Then set {x X :  (x , x) < r} is defined as
0 0
open sphere with centre x and radius r.
0
 Closed sphere:
S [x ] = {x X : (x, x ) x}
r 0 0

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