Page 216 - DMTH503_TOPOLOGY

Topology

Notes Lemma 1: 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.

Proof: Step I: Let W be open in X. We show there is a countable collection {U } of open sets of X
n
such that
W = U = U n
n
since the basis  for X is countable locally finite, we can write  =  , where each collection 
n n
is locally finite. Let  be the collection of those basis elements  such that    and B  W.
n n
Then  is locally finite, being a subcollection of  .
n n
Define U = B
n
B  n
Then U is an open set, and by Lemma “Let  be a locally finite collection of subsets of X. Then:
n
(a) Any subcollection of  is locally finite.

(b) The collection  = {A} A  of the closures of the elements of  is locally finite.


(c) U A  U A.
A  A 


U n = B
B C n

Therefore, U  W, so that
n
U  U n  W.
n
We assert that equality holds. Given x  W, there is by regularity a basis element B  such that
x B and B W. Now B   for some n. Then B   by definition, so that x  U . Thus W  U ,
n n n n
as desired.
Step II: We show that every closed set C in X is a G set in X. Given C, let W = X – C, by Step I, there

are sets U in X such that W = U n. Then
n
C = (X U n),

so that C equals a countable intersection of open sets of X.
Step III: We show X is normal. Let C and D be disjoint closed sets in X. Applying step I to the open

set X – D, we construct a countable collection {U } of open sets such that U  U n  X D.
n n
Then {U } covers C and each set U is disjoint from D. Similarly there is a countable covering
n
n
{V } of D by open sets whose closures are disjoint from C.
n
Now we are back in the situation that arose in the proof that a regular space with a countable
basis is normal. We can repeat that proof. Define
n n


U  U  i V and V  V  i U
n
n
n
n

i 1 i 1

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