The star compactification

I ntrn. J. Mh. Mah. S. Vol. 4 No. 3 (1981) 451-472 451 THE STAR COMPACTIFICATION G.D. RICHARDSON Department of Mathematics East Carolina University Greenville, N. C. 27834 D.C. KENT Department of Pure and Applied Mathematics Washington State University Pullman, Washington 99164 (Received September 18, 1980) ABSTRACT. The relationships between a convergence space and its star compactifi- cation is studied. Special attention is given to lifting properties of this compactification. In particular, it is shown that a natural extension of any continuous function to the respective compactification spaces is @-continuous. KEY WORDS AND PHRASES. Convergence space, compactification, G-space, R-series, natural extension, @-continuous function, proper map, open map. 1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 452 1. G.D. RICHARDSON and D.C. KENT INTRODUCTION. We study a convergence space compactification which was introduced by one of the authors in 1970 (see [II]). The star compactification m * (X*,i*) of a convergence space X is constructed by adjoining to X the set X’ of all non-convergent ultrafilters on X and constructing a compact convergence structure on X* X U X’ in a natural way. from a T It is proved in [II] that a continuous function space X into a compact T 2 3 space Y has a continuous extension to X*. The authors published a survey paper, [7], concerning the existence of largest and smallest convergence space compactifications relative to various constraints. In all cases studied, the largest compactification, whenever it existed, turned g:. out to be In a :re recent paper, [9], we showed that K* can be used to characterize e-regular and completely regular spaces. These results suggest that further investigation of the star compactification is appropriate. , In Section 2, we examine the relationship between the decomposition series of X and X showing that the lengths of these series can differ by at most one. These results yield a method for constructing compact T 2 spaces with arbitrarily long decomposition series. In Section 3, the R-series of X and X* are compared. By means of the R-series, the notion of e-continuity and other e-mapping properties (see [2], [3]) are extended to convergence spaces. If f is a function from a space X to a space Y, then a "natural extension" f T 2 X* / Y* is defined in Section 4. The natural extension is unique if Y is and coincides with the continuous extension constructed in [II] when Y is compact and T 3. The main result of Section 4 is that any natural extension f, is e-continuous whenever f is continuous. This result is used to obtain, among other things, an alternate construction of BX for a Tychonoff topological space X. Section 5 examines conditions on f, X, and Y under which f, is continuous, and Section 6 gives conditions under which f preserves certain quotient-type mapping 453 STAR COMPACTIFICATION properties, such as "open", "proper", and "perfect". DECOMPOSITION SERIES. 2. [7] for convergence space notation and The reader is asked to refer to terminology not given here, as well as additional information about the star As in [7], space will always mean convergence space, and the compactification. The separation axioms T abbreviation "u.f." is often used for "ultrafilter". I (singletons are closed), T 2 (convergent filters have unique limits), and T 3 (regular plus T 2) will be used, but no separation axioms are assumed unless such is explicitly stated. U(X)) be the set of all filters (resp., Given a space X, let F(X) (resp. Let X’ be the set of all non-convergent members of (X), and ultrafilters) on X. X* If A c X, define A’ X U X’. E F(X), and F’ # {F’: F } 3" then for all F * 3’; let 3 , 6 X’: A E,}, and A* { 3" If be the filter generated by then let be the filter generated by {F* :F otherwise, A U A’. 3 }. If 3’ exists, We omit the easy proofs of the first 3 two propositions. PROPOSITION 2.1. The following equalities hold for any subsets A, B of X: A’ U B’ A’ (AUB)’ PROPOSITION 2.2. (c) If (a) (X)and 6 (A UB)* Define A’ E }. generated by {A ! X If (AB)’; A* U B* F(X*), and X’ Let X be a space, (b) B’ If 6 F(X*), and F(X), 3’ exists, then (X*) and X’ 6 , ^ then 6 .’)^ 6 A* B* =(AB)*. to be the filter on X >_ *, then ^ >- 3 (X). A convergence structure is defined on X* as follows: X, For x X’ -+ x + in iff X*-, it is proved in X* iff there is + x in i* * [II] <* (X*,i*) >_ that Let X such that >- *; for denote the identity embedding of X into is a compactification of X which is T 2 G.D. RICHARDSON and D.C. KENT 454 It is immediate from the construction that, for any non- whenever X is T2. X*-X is a T2 pretopological space; thus X* compact space X, whenever X is pretopological. is pretopological The universal property of K* established in [II] will be obtained in Section 3 as a corollary of a much more general result. A subset A of space X is bounded if each ultrafilter containing A is convergent. X is said to be locally bounded if each convergent filter contains a bounded set. X is essentiall7 bounded if 3 ( X’ implies that the filters ( X’ and : % 3 } contain disjoint sets. The next proposition is proved in [7]. PROPOSITION 2.3. (a) X is locally bounded iff X is open in X*. X is essentially bounded iff X*- X is discrete. (b) We shall next consider the relationship between the closure operators of X and , cl X*. cl X be the closure operator on a space X. For an ordinal number we deflne: x A =A IA cl A X clx cl Let X clx(Cl-IA) if u-I exists A clA U < c A if is a limit ordinal. u+IA for all A c X is called the The smallest ordinal u such that clA cl X X The relationship length of the. decomposition series of X and denoted by ED(X). between ED(X) and ED(X*) can be obtained with the help of several lemmas. For the remainder of this section, we shall assume that X is an arbitrary space; (X*,i*) will always’ denote the star compactification of X. smallest infinite ordinal number. LEMMA 2.4. n If A c X, then c i x,A cl n n-I XA U (clX A)’. Let be the 455 STAR COMPACTIFICATION It suffices to prove this result for n--2. PROOF. CI2xA U (clXA)’ c__ cl2x, A x so (X*) such that CI2xA. If : (c]. If x is obvious. 8" > 2 { clx, A 0 X’ Define B / x in X and If B c X’ let { 8 X’: 8 >_ B} and B for all G (clx, B) filter + x such that + x in X. Also, 6 x for all G For each F 6 , sections of the net clx, B. >_ Thus 8 >- 8 (SF) F { B’ and x and (B~) U B’. 3 (X) { B v. / >_ If 8 in X* and B 8 >- { B’ and so containing B and a > and so 3 . we have Letting 8 B (X) such that { B such that F Then B}; note { X This implies >- (6^)’. B ", then there is 8F 3: {x By Proposition 2.2, A similar argument shows that Let 8F * B~ c__ clx, (...truncated)