Page 221 - DMTH503_TOPOLOGY

Unit 26: The Smirnov Metrization Theorem

Notes
1
Given mZ , let  be the covering of X by all these open balls of radius ; that is, let
+ m
m
ì æ 1 ö ü


 = í B C ç x, ÷ : x C and C  ý
m è
î m ø þ
Let  be a locally finite open refinement of  that covers X. (Here we use paracompactness).
m m
Let  be the union of the collections  .
m
Then  is countably locally finite.
We assert that  is a basis for X; our theorem follows.
Let x be a point of X and let be a neighbourhood of x. We seek to find an element D of  such
that xD.

Now x belongs to only finitely many elements of  say to C , …, C . Then   C is a
1 K i
neighbourhood of x in the set C , so there is an  > 0 such that
i i
B (x,) ( C ).
C i i
2
Choose m so that < min.{ , …,  }.
K
1
m
Because the collection  covers X, there must be an element D of  containing x.
m m
æ 1 ö
Because  refines  , there must be an element B y, ÷ of  , for some C and some
m m C ç è m ø m
æ 1 ö
y  C that contains D. Because xDB C ç è y, m ø ÷ , the point xC, so that C must be one of the

æ 1 ö 2
sets C , …, C . Say C = C . Since B y, ÷ has diameter at most <  , it follows that
1 K i C ç è m ø m i

æ 1 ö
xD B ç y,  B (x, ) , as desired.
i C è m ø ÷ C i i

26.2 Summary

 A space X is locally metrizable if every point x of X has a neighbourhood that is metrizable
in the subspace topology.
 A space X is metrizable iff it is a paracompact Hausdorff space that is locally metrizable.

26.3 Keywords

Hausdorff Space: A topological space X is a Hausdorff space if given any two points x, yX,
x y, there exists neighbourhoods of x, of y such that  .
x y x y
Metrizable: A topological X is metrizable if there exists a metric d on set X that induces the
topology of X.

Paracompact: A space X is paracompact if every open covering  of X has a locally finite open
refinement  that covers X.

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