Page 218 - DMTH503_TOPOLOGY

Topology

Notes Let n be fixed for the moment. Choose a neighbourhood U of x that intersects only finitely
n 0
many elements of the collection  . This means that as B ranges over  , all but finitely many of
n n
the functions f are identically equal to zero on U . Because each function f is continuous, we
n,B n n,B
can now choose a neighbourhood V of x contained in U on which each of the remaining
n 0 n
functions f for B   , varies by at most /2.
n,B n
1 
Choose such a neighbourhood V of x for each n  Z . Then choose N so that  , and define
n 0 + N 2
W = V ... V . We assert that W is the desired neighbourhood of x . Let x  W. If n  , then
1 n 0
|f (x) – f (x )|  /2
n,B n,B 0
because the function f either vanishes identically or varies by at most /2 on W. If n > N, then
n,B
|f (x) – f (x )|  Y < /2
n,B n,B 0 n
 1 
because f maps X into 0, . Therefore,
n,B  
 n 
(F(x), F(x ))  /2 < ,
0
as desired.
Step II: Now we prove the converse.
Assume X is metrizable. We know X is regular; let us show that X has a basis that is countably
locally finite.
1
Choose a metric for X. Given m, let  be the covering of X by all open balls of radius . There
m m
is an open covering  of X refining  such that  is countably locally finite. Note that each
m m m
2
element of  has diameter at most . Let  be the union of the collections  , for m   .
m m m +
Because each collection  is countably locally finite, so is . We show that  is a basis for X.
m
Given x  X and given  > 0, we show that there is an element B of  containing x that is
1 
contained in B(x, ). First choose m so that  . Then, because  covers X, we can choose an
m 2 m
2
element B of  that contains x. Since B contains x and has diameter at most  , it is contained
m
m
in B(x, ), as desired.
25.2 Summary

 A subset A of a space X is called a G set in X if it equals the intersection of a countable

collection of open subsets of X.
 Let X be a regular space with a basis  that is countably locally finite. Then X is normal,
and every closed set in X is a G set in X.

 A space X is metrizable iff X is regular and has a basis that is countably locally finite.

25.3 Keywords

Basis: A collection of subsets B of X is called a basis for a topology if:
(1) The union of the elements of B is X.
(2) If x  B B , B , B  B, then there exists a B of B such that x  B  B B .
1 2 1 2 3 3 1 2

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