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.