Page 209 - DMTH503_TOPOLOGY

Unit 24: Local Finiteness and Paracompactness

The situation where  consists of the three sets < V < W. The sets we have formed are disjoint. Notes
In fact, they are separated by a distance of at least 1/n. This means that if V and W are distinct
elements of , then d(x, y) 1/n whenever x T (V) and y T (W).
n n
To prove this fact, assume the notation has been so chosen that V < W. Since x is in T (V), then x
n
is in S (V), so the 1/n-neighbourhood of x lies in V. On the other hand since V < W and y is in
n
T (W), the definition of the latter set tells us that y is not in V. If follows that y is not in the 1/n-
n
neighbourhood of x.
The sets T ( ) are not yet the ones we want, for we do not know that they are open sets. (In fact,
n
they are closed.) So let us expand each of them slightly to obtain an open set E (). Specifically,
n
let E ( ) be the 1/3 n-neighbourhood of T ( ); that is, let E ( ) be the union of the open balls
n n n
B(x, 1/3n), for x  T ( ).
n
In case U < V < W, we have the situation. The sets we have formed are disjoint. Indeed, if V and
W are distinct elements of , we assert that d(x, y) 1/3n whenever x E (V) and y E (W); this
n n
fact follows at once from the triangle inequality. Note that for each V , the set E (V) is
n
contained in V.
Now let us define
 = {E ()|}.
n n
We claim that E is a locally finite collection of open sets that refines . The fact that E refines 
n n
comes from the fact that E (V) V for each V . The fact E is locally finite comes from the fact
n n
that for any x in X, the 1/6n - neighbourhood of x can intersect at most one element of E .
n
Of course, the collection  , will not cover X. But we assert that the collection
n
E    n
n  
does cover X.

Let x be a point of X. The collection  with which we began covers X; let us choose to be the
first element of  (in the well-ordering <) that contains x. Since is open, we can choose n so that
B (x, 1/n)  . The, by definition, x S ( ). Now because is the first element of  that contains
 n
x, the point x belongs to T ( ). Then x also belongs to the element E ( ) of E , as desired.
n n n
Self Assessment

1. Many spaces have countable bases; but no T space has a locally finite basis unless it is
1
discrete. Prove this fact.
2. Find a non-discrete space that has a countably locally finite basis but does not have a
countable basis.

24.2 Paracompactness

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

Example 2: The Space n is paracompact. Let X = n . Let  be an open covering of X. Let
B = , and for each positive integer m, let B denote the open ball of radius m centered at the
0 m
origin. Given m, choose finitely many elements of  that cover B and intersect each one with
m
the open set X B m 1 ; let this finite collection of open sets be denoted  . Then the collection

m
 =  is a refinement of . It is clearly locally finite, for the open set B intersects only finitely
m m
many elements of , namely those elements belonging to the collection  ... . Finally,
1 m
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