THE SMIRNOV COMPACTIFICATION AS A QUOTIENT SPACE OF THE STONE-CECH COMPACTIFICATION

For a separated proximity space, a decomposition of the Stone-Cech compactification is presented which produces the Smirnov compactification and its basic properties by elementary arguments without recourse to clusters or totally bounded uniformities.

ABSTRACT.For a separated proximity space, a decomposition of the Stone-Cech compacti- fication is presented which produces the Smirnov compactification and its basic properties by elementary arguments without recourse to clusters or totally bounded uniformities.
KEY WORDS AND PHRASES.Proximity space, compactification, quotient space.

I. INTRODUCTION.
It has long been recognised that (i) every T 2 compactification of a T31/2 topological space can be obtained as a quotient space of its Stone-ech extension, and (ii) every (separated) proximity space can be densely embedded in a compact proximity space, its Smirnov compactification; see, for example, [I] and [2].The purpose of this note is to present an explicit construction whereby the Smirnov compactification can, as is implicit in the above results, be derived from the Stone-6ech.Since it is markedly simpler than the construc- tions usually employed, the procedure has pedagogical utility in addition to its intrinsic interest; the author has found it to be of considerable convenience in giving a brief introduction to proximity space theory to final year undergraduates who had completed a course in general topology.

CONSTRUCTION.
Given a separated proximity space (X,6), with associated T3 topological space (X,(6)) regarded as a (topological) subspace of its Stone-ech compactification X, let S and int(S) denote the closure and interior in the space X of a subset S (of X or of BX).Recall the notation A<< B to mean A X B (for subsets A, B of X).The construc- tion proceeds by identifying points of X whenever they are indistinguishable (in a natural sense) from" within (X,6).We begin by observing the following result, generally obtained as a consequence of the Smirnov compactlfication (see, for example, [2,Theorem 7.12]), but which to avoid circularity can be obtained by an argument like that which establishes Urysohn's lemma.
LEMMA i.If A B then there is a continuous mapping f: X [0,i] taking the values 0 and throughout A and B, respectively.
PROPOSITION i.The binary relation % defined on 8X thus: p % q if and only if there do not exist subsets A,B of X such that p e A, q e B, A B is an equivalence relation.
PROOF.Reflexivity follows from Lemma since the continuous extension over 8X of such an f will map A and B to 0 and i, implying A N B .
Symmetry is immediate.For transitivity, suppose if possible that p % q, q % r and p r, and choose subsets A,C and B of X so that p e A, r e C, A C, A B, X B C.
Since q e B U X B this contradicts either p % q or q % r.Now for each p e 8X denote by 8(p) the equivalence class containing p, and by aX the set of all these equivalence classes, so that 8 becomes a mapping from 8X onto aX.
Gie X the quotient topology induced by 8, and we have immediately that 8 is continuous, X is compact, 8(X) is dense in oX. (2.1) In the investigation of this quotient space it will be helpful to know that 8 is closed mapping and thus the decomposition is upper semi-continuous, which is the point of Lemma 4 below.We first establish an alternative characterization (Lemma 3) of the relation %.
LEMMA 2. If A B in (X,6) then A c int(B) in 8X.
PROOF.This is almost immediate from Lemma i.
LEMMA 3.For p, q e 8X, p q if and only if there are neighbourhoods Np of p, Nq of q (in 8X) such that N X N 0 X.P q PROOF.If such neighbourhoods exist then p e N X and q e N X, hence p q. P q Conversely if p q choose A, B c X so that p e , q e and A B. Using [2, Cot.
3.5 and Lemma 2.8] we may find closed subsets C,D of X such that A C, B D and C D: then Lemma 2 shows that C and D are neighbourhoods of p and q whose traces on X are not 6-related.
-i LEMMA 4. Let A be a closed subset of 8X; then so is 8 (8(A)).
- A has a finite subcover, say {Na(1), Na( 2) Na(n)}; and the neighbourhood U a(i) Ua(2) -I(8(A)) in at least one point v, where v % a' for .. 0 U of u must intersect 8 a(n) some a' e A. Then (for some j between and n) a' Na(j), so that Ua(j) and A(j) are neighbourhoods of v and a', repectively, whose traces on X are not 6-related, giving the contradiction v a'.
Standard quotient-space results obtain from Lenna 4 the following, where cl denotes closure in the space oX: 8 is closed, oX is T 2, and for each subset A of X we have 8(A) cl(8(A)). (2.2) Being a compact T 2 space by (2.1) and ( 2.2), oX possesses a unique compatible proximity, the relation A between its subsets given by C A D if and only if el(C) n el(D) .
It remains to examine the way in which 8 embeds (X,6) into (oX,A), beginning with the following observation which establishes that 8 acts injectively on X: LEMMA 5.For each x e X, 8(x) {x}.
PROOF.Consider any z in 8X distinct from x.If we choose a closed neighbourhood Z of z not including x, then X N (SX Z) is an open neighbourhood in X of x, so {x} X (X n (SX Z)) X N Z.Since x e {x} and z e X 0 Z this gives x z.
The final verificational step in the construction is to check that 8 is a proximity- isomorphism between (X,6) and the proximity subspace 8(X) of (oX,A): PROPOSITION 2. For subsets A, B of X, A 6 B if and only if cl(8(A)) cl(8(B)) .
PROOF.If there exists y in cl(8(A)) cl(8(B)) then (2.2) shows that we can find p e A, q e B such that y 8(p) 8(q); and since p q we get A 6 B.
Conversely, suppose that A 6 B. We observe that the family of sets {A N C C B} possesses the finite intersection property, whence the compactness of 8X guarantees that it contains a point p which is common to their closures.For each neighbourhood N of p in 8X, N 0 X 6 B (since otherwise B X N, and the choice of p yields a contradiction).
It follows that the family {B 0 M M N 0 X, N a variable neighbourhood of p} also possesses the finite intersection property.A second appeal to compactness produces q e 8X common to their closures.Thus each neighbourhood of q meets every such set B M. (2.3) Now if p,q were not %-related we would be able to find neighbourhoods P,Q (respectively) of them such that P N X Q X; however, this gives us X Q >> P 0 X from which (2.3) produces the contradiction that Q intersects B N (X Q).Hence p % q i.e. 8(p) 8(q).Since p e A and (via (2.3)) q e B we now see using (2.2) that ep; e(A) n O(B) cl(e(A)) n cl(O(B)).
Snmarizing, we have seen that oX is a compact (separated) proximity space possessing a dense subspace which is isomorphic to X; that is, TKFOREM.(oX, A) is the Smirnov compactification of (X,6).
NOTE.The above procedure, in addition to constructing the Smirnov compactification, provides a convenient base from which to establish its fundamental properties.For example, let there be given a proximity mapping f from (X,6) into a compact separated proximity space (Y,6'); and denote by f* the continuous extension of f over 8X.It is routine to confirm that the formula f(e(x)) f*(x) gives a well-defined and continuous mapping fo from oX to Y, so f has a proximity mapping extension over X.The essential uniqueness of the Smirnov compactification can be proved merely by checking that if (Z,6") is any compact separable proximity space containing X as a dense subspace then the extension over oX of the inclusion of X in E is injective; and virtually the same argument shows that, given a T 2 compactification 7X of a topological space X, the Smirnov compactification of X under the proximity induced by X is indistinguishable from X itself: whence the one-to-one correspondence between compatible proximities and T 2 compactifications follows.