8 PETEK H. KRAUSS AND DAVID M. CLARK
F % define
h*(F) = {U J I F U)
and l e t
ft* = {h*(F) | F R } .
Then IR,* is a ring of subsets of J and h*:?£-» TR* is an isomorphism.
Since 0 32, 3& is a basis of closed sets for a topology S'(TZ) on I
and ft* is a basis of closed sets for a topology 3"(3^*) on J.
Lemma 1.1 J is a compact TQ-space. 1
Lemma 1.2 For every ¥ £ H,
9
(i) h(F) = h(l) 0 h*(F)
(ii) h(l - F) = h(l) - h*(F). I
Corollary 1.3 h(l) f l J is a dense subset of J. I
Lemma 1.U If 3^, covers I then the following are equivalent:
(i) I is a T -space.
(ii) h is one-to-one.
(iii) h is a homeomorphism of I onto h(l) (with the relative topo-
logy). I
If ft, covers I and I is a T -space then J is called the Wallman
T -compactification of I.
Lemma 1.5 I is compact if and only if M c h(l). I
Corollary 1.6 If h(l) = J then I is compact. 1
Lemma 1.7 Every open subset of I is the union of closed sets if and
only if h(l)c M. I
Notice, if I is a T -space then every (open) subset of I is the
union of closed sets, and if every open subset of I is the union of closed
sets then V, covers I.
Previous Page Next Page