Compactiﬁcation and compactoidiﬁcation

. After discussing some of the many ways to get the Banaschewski compactifi-cation 03B20T of an arbitrary ultraregular space T, we develop another construction of 03B20T in Th. 2.1. Using those ideas, we develop an analog of 03B20T2014what we call a compactoidification K,T of an ultraregular space T in Sec. 3; r~T is, in essence, a complete absolutely convex compactoid ’superset’ of T to which continuous maps of T with precompact range into any complete absolutely convex compactoid subset may be ’continuously extended.’


The Many Faces
For any topological spaces X and Y, C (X, Y) and C* (X, V) denote the spaces of continuous maps of X into Y and the continuous maps of X into Y with relatively compact range, respectively. To say that a topological space X is ultraregular or ultranormal means, respectively, that the clopen sets are a basis or disjoint closed subsets of X may be separated by clopen sets. A synonym for ultraregular is 0-dimensional. We have a slight preference for the former in order to avoid confusion with other notions of dimension. Throughout the discussion, T denotes at least a Hausdorff space. For an ultraregular space E containing at least two points and ultraregular T, B. Banaschewski [2] discovered a compactification 03B20T of T in which every x E C* (T, E) may be continuously extended to 03B20x E E). 03B20T is nowadays usually called the Banaschewski compactification of T. It functions as the natural analog of the Stone-ech compactification (03B20T is aT for ultranormal T) in non-Archimedean analysis. Like the Stone-Cech compactification, the Banaschewski compactification is a protean entity, assuming many different guises. We discuss some of them in this section and then develop a new one in Sec. ?.

1.1
As a completion Let E be an ultraregular space containing at least two points and let T be ultraregular. Let C* (T, E) denote the weakest uniform structure on T making each x E C* (T, E) uniformly continuous into the compact space cl x (T) equipped with its unique compatible uniform structure. By [I], pp. 92-93, since T is ultraregular, C* (T, E) is compatible with the topology on T and C* (T, E) is a precompact uniform structure on T. Since C* (T, E) is precompact, its completion 03B20T is compact and is called the Banaschewski compactification of T. 03B20T is ultranormal ( [2], p. 131, Satz 2 or [1], p. 93, Theorem I)-hence ultraregular-and, by the usual process of extension by continuity function from a dense subspace to the whole space, each x E C* (T, E) may be continuously extended to a unique continuous function 03B20x E C* E). 03B20T is unique in a sense we discuss in the context of E-compactifications (Th. 1.6). At this point the reader may find the notation /3oT curious. Why 03B20T and not ,QET?
As long as E is ultraregular and contains at least two points ([I], p. 93, ~8~, pp. 240-243). the uniformity C* (T, E) does not depend on E! A fundamental system of entourages for C* (T, E), no matter what E is, is defined by the sets where P is any finite open (therefore clopen) cover of T by pairwise disjoint sets. The completion of T with respect to this uniformity is the way Banaschewski obtained 03B20T. The definition of PoT as the completion of C* (T, E) where E is the discrete space of integers was first given in [7], though the idea of treating compactifications as completions is due to Nachbin. The connection with the Stone-Cech compactification is the following. Definition 1.1 Let P be a finite clopen cover of a topological space S by pairwise disjoint sets and let V denote the uniformity generated by Vp . YYe say that S is strongly ultraregular . Theorem 1.2 ([8], pp. 251-2) (a) Every ultranormal T1-space S is strongly ultraregular.
(b) If a topological space S is strongly ultraregular then 03B20S = 03B2S.

As an E-Compactification
Tihonov proved that a completely regular space T may be characterized as one that is homeomorphic to a subspace of a product [0, 11m of unit intervals. Even though his name is not associated with it, he created the first version of the Stone-ech compactification /3T of T by then taking the closure of T in [0, 1]m. Engelking and Mrówka [5] developed analogous notions of E-completely regular space T and E-compactification 03B2ET. Let Sand E be two topological spaces. S is called E-completely regular if it is homeomorphic to a subspace of the m-fold topological product Em for some cardinal m. If E = R or [0, 1] , this is the familiar notion of complete regularity. With 2 denoting the discrete space {0,1 }, it happens that An E-compact space is one which is homeomorphic to a closed subspace of a topological product Em for some cardinal m. The 2-compact spaces are characterized as follows: (b) The space 03B2ET is unique in the sense that if S is an E-compact space containing T as a dense subset and such that every continuous x : T ~ E has a continuous extension to S, then S is homeomorphic to 03B2ET under a homeomorphism that is the identity on T.
(c ) T is E-compact if and only if T = 03B2ET.
How does this apply to 03B20T? Ultraregular spaces Tare 2-completely regular by Th. 1.3. Since 03B20T is compact and ultranormal, it follows that /3oT is 2-compact by Th. 1.4. Therefore, by Th. 1.5(b) it follows that Theorem 1.6 UNIQUENESS OF PoT is homeomorphic to ~32T under a homeomorphism that is the identity on T, as would be any ultraregular compactification of an ultraregular T with the E-extension property.

As a Space of Characters
Let F be an ultraregular Hausdorff topological field so that X = C. (T, F) may be considered as an F-algebra. A character of X is a nonzero algebra homomorphism from X into F. Let the set H of characters of X be equipped with the weakest topology for which the maps H --~ F, h -h (x), are continuous for each x E C* (T, F). For each p E /3oT let pA denote the evaluation map at p, the map C' (T, F) ~ F, x ~ 03B20x (p). It is trivial to verify that each p' is a character of C. (T, F) . But more is true: You get all the characters of C. (

Characters Again
Once again 03B20T is realized as a space of nonzero homomorphisms-ring homomorphisms this time-into the very simple (discrete) field 2 with 2 elements.
A commutative ring X with identity in which each element is idempotent is called a Boolean ring. A subcollection X of the set of subsets of a given set T which is closed under union, intersection and set difference of any two of its members is called a ring of sets.
Such a collection forms a ring in the usual algebraic sense if addition and multiplication are taken to be symmetric difference and intersection, respectively. If the sets in X cover T then X is called a covering ring. Since X must have a multiplicative identity (i.e., with respect to intersection) any covering ring must contain T as an element. Any covering ring X generates (in the sense that it is a subbase for) a ultraregular topology on T ; the topology is ultraregular since the complement T -A of any open set (member of X) must belong to X. In the converse direction, the class CI (T) of clopen subsets obviously constitutes a covering ring of any topological space T.
Let X be a Boolean ring and endow 2x with the product topology. The Stone space S (X) of the Boolean ring X is the subspace of 2X of all nonzero ring homomorphisms of X into 2. S (X ) is called the Stone space because of Stone's use of it in his remarkable characterization of compact ultraregular spaces. THE STONE REPRESENTATION THEOREM ( [12], Theorem 4, [12], [4] p.227 or [6j, pp. 77-80) If T is a compact ultraregular space, then T is homeomorphic to the Stone space of the Boolean ring CI (T) of clopen subsets of T. Conversely, the Stone space S (X ) of any Boolean ring X is a compact ultraregular Hausdorff space and X is ring-isomorphic to the Boolean ring CI (T) of clopen subsets of S (X ).
If T is ultraregular then 03B20T is the Stone space of CI (T). Indeed, the map p : T -+ S (CI (T)), t ~ 03B2t, defined for t ~ T and K ~ CI (T) by 1 ~ 2 t ~ K (03B2t)(K) = {0 ~ 2 t ~ K is a homeomorphism of T onto a dense subset of the compact ultraregular Hausdorff space S (Cl (T)). .

As a Space of Measures
Let T be ultraregular and let Cl (T) be the ring (algebra, actually, since T E CI (T )) of clopen subsets of T, and let F be an ultraregular Hausdorff topological field. A 0-1 measure on T is a finitely additive set function m : CI (T) --~ ~0,1) C F satisfying the condition: m (U) = 0 and in other words, that clopen subsets of sets of measure 0 also have measure 0. Measures mt 'concentrated at points t E T' (also called 'purely atomic' or 'the point mass at t')) which where the Sj are clopen sets and n E N. It is trivial to verify that the map t -mt is a homeomorphism of T into M. Using the techniques of (IJ one can demonstrate that M is a compact ultranormal Hausdorff space to which any x E C* (T, F) may be continuously extended. It follows that 03B20T = M in the sense of Th. 1.6.
Last, let us mention that 03B20T may also be realized as a wallman compactification utilizing the lattice of clopen subsets of T..

A New Approach
A construction of 03B20T using the methods of non-Archimedean functional analysis is presented in Theorem 2.1. The proof hinges on the fact that, for a local field F, if U is a neighborhood of 0 in a locally F-convex space X then its polar U° is Q (X', X)-compact ( [15], Th. 4.11).
Theorem 2.1 Let F be a local field, let T be ultraregular and let C'(T, F) denote the supnormed space of all continuous F-valued functions on T with relatively compact range. There is an ultranormal compactification 03B20T of T such that any x E C*(T, F) may be continuously extended to a function 03B20x E C (03B20T, F).
Proof. For t E T, let t" denote the evaluation map x ~----~ x(t) for any x E C'(T, F). ~Ve note that each such is a continuous linear form (algebra homomorphism, actually) and is of norm one. Thus T " = {t^ : t E T} C U where U denotes the unit ball of the normdual C*(T, F)' of C*(T, F). Furthermore, the map i T ~ C*(T, Fl', t ~ t", embeds T homeomorphically in C*(T, F)' endowed with its weak-* topology by the following argument. The map i is obviously injective. If a net t~ --~ t E T then x(t,) -~ x(t) for any x E C*(T, F); hence -; t" and therefore i is continuous. To see that i is a homeomorphism onto let ~'1 be a closed subset of T. Since T is ultraregular, if t ~ I( then there exists x E C*(T, F) such that x(t) = 0 and |x(K)| = r > 1. Hence the polar of tx} is a neighborhood of t" disjoint from and is a closed subset of i(K). As U is the polar of the unit ball of C*(T, F), it follows that U is weak-*-compact ([IS), Th. 4.11}. Therefore the closure cT in U of (the homeomorphic image of ) T " is compact in C*(T, F)' endowed with the weak-* topology. As to the continuous extendiblity of x E C'(T, F), consider the canonical image Jx of x in the second algebraic dual of C*(T, F), i.e., for any f E C*(T, F)', Jx(f) = f (x). Clearly Jx is weak-*-continuous on C*(T,F)'; so, therefore, is its restriction Qoz = Jx Should this be called cFT rather than cT? No topologically significant changes occur for different F's: the compactness of the ultraregular space cT and the fact that T is C*-embedded in cT imply that cT = 03B20T by Th. 1.6.

Compactoidification
In this section we construct a compactoidification r~T of an ultraregular space T. (F, ~~!) denotes a complete nontrivially ultravalued field throughout. As usual, we abbreviate 'Fconvex' to 'convex.' A map f defined on an absolutely convex subset A of a vector space over F with values in some absolutely convex set in a vector space over F is called affine if f (ax + by) = a f (x) + b f (y) for all x, y E A and all a, b E F with ~a~ I and ~b~ 1. Definition 3.1 A compactoidification of an ultraregular space T is a pair (i, 03BAT) where 03BAT is a complete absolutely convex compactoid subset of some Hausdorff locally convex space E over F and t : T --> KT is a continuous map with precompact range for which following extendibility property holds : For any complete absolutely convex compactoid subset A of some Hausdorff locally convex space E over F and any continuous map j : T -~ A with precompact range, there exists a unique continuous affine map J : KT --~ A such that J o i = j. Since the identity map II : t ~ t of 03BA1T onto KIT also satisfies II o il = ii, it follows from the uniqueness that fi = Ji o Jz. Similarly, 12 = J2 o Ji where 12 is the identity map of K2T onto K2T. It follows that Ji is a homeomorphism of 03BA1T onto K.2T and J2 is its inverse. if ii (ti) = ii (t2) then i2 (ti) = Ji o ii (tl) = J1 o il (t2) = i2 (t2) so if one of the maps i is 1-1, all such i must be. As shown in Theorem 3.3, there is an i that is 1-1.
In the notation of Sec. 2: 3.1), so, therefore, is the closed absolutely convex hull ~T of the compact set cl T" . . It follows from [10], Prop. 1.3 that B is edged (i.e., if the valuation of F is dense then cl B = n {a( clB) : a E F, (ai > 1}) and therefore ([9], Th. 4.7) a polar set in C" (T, F)'. If cl B ~ U there must exist g E C* (T, F)" such that (g~ 1 on Band 19 ( f )~ > 1 for some f E U-cl B. Since g must be an evaluation map determined by some point x E C* (T, F) by [9], Lemma 7.1, we have found an 2 such that ( = (~) ~ 1 for all t E T but f (~)~ > I. As this contradicts ~j 1, the proof of (a) is complete. (b) As in the proof of Th. 2.1, i is a homeomorphism onto the precompact set T ". To verify the extendibility requirement, let A be a complete absolutely convex compactoid and let j : T ~ A be continuous with precompact range. V'e define the affine extension J of j on the absolutely convex hull B of T^ by taking J 1 a=ti") _ aij (t=) for a= e F, |ai| I 1, i = 1, ... , n. The definition makes sense because the t;" are linearly independent for distinct ti. Evidently j = J o i. To prove the continuity of J, let s -s = asitsi^ be a net in B convergent to 0 in the weak-* topology. Let [A] denote the linear span of A and note that for any f E ~A~', the map f o j E C* (T, F) ,since j (T) is precompact. Thus, f(J( s)) = f (03A3 03B1sij ( t s i ) ) = 03A3 a s i f ( j ( t s i ) ) = s (f o j) ~ 0 and we conclude that J (~~) --> 0 in the weak topology of (Aj . As A is of countable type. hence a polar space, the weak topology coincides with the initial one on the compactoid A ([9], Th. 5.12) so J (~cs) -~ 0 in A. By continuity and 'affinity,' J extends uniquely to a continuous affine map of cl B = r~T into A, since A is complete.