Realcompact Research Articles

A Tychonoff Almost Realcompactification

Let $X$ be a Tychonoff topological space. A Tychonoff almost realcompact space $aX$ is constructed that contains $X$ as a dense subspace and has the property that if $f:X \to Y$ is continuous and $Y$ is Tychonoff and almost realcompact, then $f$ can be extended continuously to $aX$. Several characterizations of $aX$ are given, and the relationships between $aX$, the Hewitt realcompactification $vX$, and the minimal $c$-realcompactification $uX$ are investigated. Properties of the projective covers of these spaces, and their relation to $vE(X)(E(X)$ denotes the projective cover of $X$), are discussed.

Wallman-type compactifications on 0-dimensional spaces

Let E E be Hausdorff 0 0 -dimensional, D \mathcal {D} the discrete space { 0 , 1 } \{0, 1\} , and N \mathcal {N} the discrete space of all nonnegative integers. Every E E -completely regular space X X has a clopen normal base F \mathcal {F} with X ∖ F ∈ F X\backslash F \in \mathcal {F} for each F ∈ F F \in \mathcal {F} . The Wallman compactification ω ( F ) \omega (\mathcal {F}) is D \mathcal {D} -compact. Moreover, if an E E -completely regular space X X has a countably productive clopen normal base F \mathcal {F} with X ∖ F ∈ F X\backslash F \in \mathcal {F} for each F ∈ F F \in \mathcal {F} , then the Wallman space η ( F ) \eta (\mathcal {F}) is N \mathcal {N} -compact. Hence, if X X has such an F \mathcal {F} , and is an F \mathcal {F} -realcompact space, then X X is N \mathcal {N} -compact.

A Tychonoff almost realcompactification

Let Xbe a Tychonoff topological space. A Tychonoff almost realcompact space aX is constructed that contains X as a dense subspace and has the property that if f: XY is continuous and Y is Tychonoff and almost realcompact, thenf can be extended continuously to aX. Several characterizations of aX are given, and the relationships between aX, the Hewitt realcompactification vX, and the minimal c-realcompactification uX are investigated. Properties of the projective covers of these spaces, and their relation to vE(X) (E(X) denotes the projective cover of X), are discussed.

Certain Subsets of Products of θ-Refinable Spaces are Realcompact

Certain Subsets of Products of θ-Refinable Spaces are Realcompact

Prabir Roy's space Δ is not n-compact

P. Roy introduced the space Δ as an example of a metric space satisfying ind X−0, dim X >. It is shown here that Δ cannot be embedded as a closed subspace in a product of discrete spaces. This provides a counterexample to a long-standing conjecture that every zero-dimensional realcompact space admits such an embedding. Related problems of dimension theory are also discussed.

Wallman compactifications and Wallman realcompactifications

In 1964 Frink [8] generalized Wallman's method [17] of compactification and asked the question: “Is every Hausdorff compactification of a Tychonoff space a Wallman compactification?”. This problem, which is as yet unsolved, has led to the discovery of a number of necessary and/or sufficient conditions for a Hausdorff compactification to be Wallman; (see Alò and Shapiro [2], [3], Banasĉhewski [6], Njåstad [13] and Steiner [15]). Recently Alò and Shapiro [4], [5] have generalized the Wallman procedure to discuss what they call Z*-realcompact spaces and Z*-realcompactification n(Z) corresponding to a countably productive (c.p.) normal base Z on X. Some further work has been done by Steiner and Steiner [14].

Realcompactifications of Products of Ordered Spaces

The equality $\upsilon (X \times Y) = \upsilon X \times \upsilon Y$ is studied for the case when one of the factors is a linearly ordered topological space (LOTS). Among the results obtained are the following: 1. If $X$ is any separable realcompact space and $Y$ is any LOTS of nonmeasurable cardinal, then $\upsilon (X \times Y) = \upsilon X \times \upsilon Y$. 2. If $X$ is a nonparacompact LOTS, then there is a paracompact LOTS $Y$ such that $\upsilon (X \times Y) \ne \upsilon X \times \upsilon Y$. 3. For any pair $X,Y$ of well-ordered spaces, $\upsilon (X \times Y) = \upsilon X \times \upsilon Y$.

On maps which are perfect with respect to the Hewitt realcompact extension

On maps which are perfect with respect to the Hewitt realcompact extension

Realcompactifications of products of ordered spaces

The equality υ ( X × Y ) = υ X × υ Y \upsilon (X \times Y) = \upsilon X \times \upsilon Y is studied for the case when one of the factors is a linearly ordered topological space (LOTS). Among the results obtained are the following: 1. If X X is any separable realcompact space and Y Y is any LOTS of nonmeasurable cardinal, then υ ( X × Y ) = υ X × υ Y \upsilon (X \times Y) = \upsilon X \times \upsilon Y . 2. If X X is a nonparacompact LOTS, then there is a paracompact LOTS Y Y such that υ ( X × Y ) ≠ υ X × υ Y \upsilon (X \times Y) \ne \upsilon X \times \upsilon Y . 3. For any pair X , Y X,Y of well-ordered spaces, υ ( X × Y ) = υ X × υ Y \upsilon (X \times Y) = \upsilon X \times \upsilon Y .

Local Topological Properties and One Point Extensions

In 1957, Mrowka [12] showed that a locally paracompact space admits a one point paracompactification (see also [2, Chapter 9, § 4, Exercise 27]). Similarly, in [9] Isiwata obtained a one point realcompactincation for locally realcompact spaces. Recently a number of authors (see [11, 16; 17; 18; 21]) have constructed one point P-extensions of local P-spaces for a variety of topological properties P. It is the purpose of this paper to draw together the various techniques used by the above mentioned authors and to study the set (lattice) of all one point P-extensions of a particular space.

An embedding characterization of almost realcompact spaces

Any Hausdorff space X can be embedded into the product space $\Pi \{ R_f^ + :f \in {C_ + }(X)\}$, where ${R^ + }$ is the set of all nonnegative reals with the topology consisting of ${R^ + }$ and all sets of the form $\{ x \in {R^ + }:x < a\} ,a \in R$, and ${C_ + }(X)$ is the set of all continuous functions from X to ${R^ + }$. Almost realcompact Hausdorff spaces are characterized as maximal Hausdorff subspaces in their closures in the product.

Supports of continuous functions

Gillman and Jerison have shown that when X is a realcompact space, every function in C ( X ) C(X) that belongs to all the free maximal ideals has compact support. A space with the latter property will be called μ \mu -compact. In this paper we give several characterizations of μ \mu -compact spaces and also introduce and study a related class of spaces, the ψ \psi -compact spaces; these are spaces X with the property that every function in C ( X ) C(X) with pseudocompact support has compact support. It is shown that every realcompact space is ψ \psi -compact and every ψ \psi -compact space is μ \mu -compact. A family F \mathcal {F} of subsets of a space X is said to be stable if every function in C ( X ) C(X) is bounded on some member of F \mathcal {F} . We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space.

Not every 0-dimensional realcompact space is 𝑁-compact

Not every 0-dimensional realcompact space is 𝑁-compact

Mappings and realcompact spaces

The problem of preserving realcompactness under perfect and closed maps is studied. The main result is that realcompactness is preserved under closed maps if the range is a normal, weak cb, A;-space. Generalizing a result of Frollk, we show that realcompactness is preserved under perfect maps if the range is weak cb. Moreover, the problem of preserving realcompactness under perfect maps may be reduced to the following: When does the absolute of X being realcompact imply that X is realcompact? Likewise the problem of preserving topological completeness under perfect maps may be reduced to an analogous question. The following special case is also proved. If φ is a closed map from X onto a weak cb, g-space Y, then X is realcompact implies that Y is realcompact. A function is a fc-covering map if any compact set in the image space is contained in the image of a compact set in the domain. We show that all closed maps whose domain X is topologically complete are λ -covering maps. Next a representation is obtained for a closed image Y of a topologically complete, G^-space. Namely Y= Ya U (U Yd where φ~\y) is compact for all yeYa and each Y{ is discrete in Y. This generalizes a theorem of K. Morita and is similar to a theorem of ArhangePskiί which replaces topologically complete with point-paracompact. If φ is an open-closed mapping from a realcompact space X onto a A>space Y, then dφ-^y) is compact for every yeY. It then follows from a result of Isiwata's that Y is also realcompact. Now let φ be a WZ-m&p of X onto a realcompact space Y. We show that X is realcompact if and only if cl^ x Φ~\y) — Φ~\y) This generalizes Isiwata's result that X is realcompact if Φ~ι{y) is a C*-embedded, realcompact subset of X. The reader is referred to [6] for the basic ideas of rings of continuous functions. The following characterization of realcompactness will be used in this paper. A completely regular, Hausdorff space X is realcompact if and only if for each p in βX — X there exists an / in C{βX) such that f(p) = 0 and f(x) > 0 if x is in X. A map will be used to designate a continuous onto function. A map is said to be closed (open) if the image of each closed (open) subset of the domain is closed (open) in the range. A map is called a Z-map if the image of each zero-set in X is closed in Y. If the inverse image of each compact set in the range is compact, then the map is said to be compact. Perfect maps are those which are closed and compact.

The α-Closure αX of a Topological Space X

Introduction. It is known that every completely regular space S has a real compactification vX, contained in ,BX, with the following property: every fE C(X) has an extension fE C(vX) [2, p. 1 18 ]. (A space S is real-compact iff every Z-ultrafilter with the countable intersection property is fixed. A real compactification of X is a real compact space in which S is densely imbedded.) This paper is a study of a-spaces (every open ultrafilter with the countable intersection property converges). The main results are contained in ?III. We show that for any space X, there exists an a-closure (see Definition 3.7) aX of X, contained in KX [3, p. 89], with the following property: If Y is any other a-closure of S and i: X-* Y is the inclusion, then i can be extended continuously to a function 1: aX-* Y. Since the construction of aX is based on KX and the structure of KX is related to open ultrafilter, therefore in ?I, we study open ultrafilters and in ?II, we state a main theorem about KX. The author wishes to thank the referee for his suggestions which made the paper more concise.

On the Hewitt realcompactification of a product space

One of the themes of Edwin Hewitt's fundamental and stimulating work [16] is that the Q-spaces (now called realcompact spaces) introduced there, although they are not in general compact, enjoy many attributes similar to those possessed by compact spaces; and that the canonical realcompactification vX associated with a given completely regular Hausdorff space X bears much the same relation to the ring C(X) of real-valued continuous functions on X as does the Stone-Cech compactification gX to the ring C*(X) of bounded elements of C(X). Much of the Gillman-Jerison textbook [12] may be considered to be an amplification of this point. Over and over again the reader is treated to a theorem and then, some pages later, to its analogue. One of the most elegant /3 theorems whose analogue remains unproved and even unstated is the following, given by Irving Glicksberg in [13] and reproved later by another method in [11] by Zdenek Frolik: For infinite spaces X and Y, the relation 3(X x Y) = gX x / Y holds if and only if X x Y is pseudocompact. The present paper is an outgrowth of the author's unsuccessful attempt to characterize those pairs of spaces (X, Y) for which v(X x Y) = vX x v Y. It is shown (Theorem 2.4) that, barring the existence of measurable cardinals, the relation holds whenever Y is a k-space and vX is locally compact; and more generally (Theorem 4.5) that the relation holds whenever the k-space Y and the locally compact space vX admit no compact subsets of measurable cardinal. The question arises naturally as to when it will occur that vX is locally compact. Our best result in this direction, a part of Theorem 4.8, is, again, not definitive: X is locally pseudocompact if and only if there is a locally compact space Y for which Xc Yc vX. Some of the questions treated here are susceptible to attack when thrown into the uniform space context. See Onuchic [18] and, more intensively, Isbell [17, especially Chapter 8]. I am indebted to Tony Hager for several references, and for allowing me to announce here that, in addition to the results of [15], he has achieved certain new theorems on the relation v(X x Y) = vX x v Y, which being uniform and not topological have negligible overlap with those of this paper.

Baire sets in topological spaces

In this paper we study topological properties of Baire sets in various classes of spaces. The main results state that a Baire set in a realcompact space is realcompact; a Baire set in a topologically complete space is topologically complete; and that a pseudocompact Baire set in any topological space is a zero-set. As a consequence, we obtain new characterizations of realcompact and pseudocompact spaces in terms of Baire sets of their Stone-Cech compactifications. (Lorch in [3] using a different method has obtained either implicitly or explicitly the same results concerning Baire sets in realcompact spaces.) The basic tools used for these proofs are first, the notions of anr-compactification andr-embedding (see below for definitions), which have also been defined and used independently byMrowka in [4]; second, the idea included in the proof of the theorem: “Every compact Baire set is aG δ ” as given inHalmos' text on measure theory [2; Section 51, theorem D].

The Dedekind completion ofC(𝒳)

The question to which this study addresses itself is the following: given a completely regular space <%f, is the Dedekind completion of G{<%f) isomorphic to C(g^) for some space ^/η. Here, C(J^) denotes the ring of continuous realvalued functions on <%f under pointwise order. Affirmative answers were provided by Dilworth for the class of compact spaces in 1950 and by Weinberg for the class of countably paracompact and normal spaces in 1960. It remained an open question whether there were any spaces for which a negative answer held. In this paper, we provide a necessary and sufficient condition that the Dedekind completion of Cί^ 7 ), for ^ a realcompact space, be isomorphic to C(^/) for some ^/. Using this, we are able to provide an example of a space gf for which the Dedekind completion of G(<%?) is not isomorphic to C{^/) for any space f. Specifically, we define and characterize a class of spaces which we call weak cb-spaces: those spaces J2f with the property that every locally bounded, lower semicontinuous function on ^ is bounded above by a continuous function. We then prove that for an arbitrary (completely regular) space J2f, the Dedekind completion of C{^f) is isomorphic to some C(^/) if and only if v^7 (the Hewitt realcompactification of g?) is a weak cδ-space. The sufficiency of this condition actually generalizes Weinberg's result, as is shown by examples; the necessity provides the negative result referred to above. The preliminary investigation of the Dedekind completion is done in Section 1, in the setting of an arbitrary (^-algebra. In Section 2, we study the connection between the lattice of normal upper semicontinuous functions on a completely regular space ^ and the minimal projective extension of gf'. This leads to the observation, in Section 4, that for a weak c6-space g? the Dedekind completion of C{^f) is isomorphic to C(3/)y where ^/ is the minimal projective extension of gf. Weak cδ-spaces are studied in Section 3, and Section 4 contains our main result. It is a pleasure to acknowledge fruitful conversations and correspondence on the subject of this paper with G. A. Jensen, J. E. Kist, and E. C. Weinberg. Their contributions and encouragement are sincerely appreciated.

The intersection of the free maximal ideals in a complete space

Introduction. In [1, p. 123] Gillman aiid Jerison prove that if X is realcompact, the intersection of the free maximal ideals in C(X) is CK(X), the ring of functions with compact support. Other authors [2], [3] have proved this result for discrete spaces and p-spaces, respectively. This note provides a proof of the fact that the result holds for any space admitting a complete uniform structure. We point out also that unlike that of the Gillman-Jerison theorem, our proof requires no prior construction of ,BX, the Stone-Cech compactification of X. Since a realcompact space is complete in the structure generated by C(X), our result extends the Gillman-Jerison theorem. (Whether every complete space is realcompact is an unsettled question: its resolution would require a proof of the existence or nonexistence of measurable cardinals.) We will employ the same terminology as in [1] and assume all spaces are completely regular and all uniform structures Hausdorff. Also, we take our uniform structures to be defined by pseudometrics. For a topological space X, C(X) denotes its ring of real valued continuous functions, and for f E C(X), Zf = { x: f(x) = 0 1. Thus, the support of f is clx [X-Zf ] and CK(X) is the ring of functions with compact support. An ideal IC C(X) is free provided nf {Zf:f I} = 0 and a maximal ideal which is free is called a free maximal ideal. A subset SCX is C-embedded in X provided each fE C(S) admits a continuous extension to all of X.

Remarks on real-compact spaces

Remarks on real-compact spaces