Minimal Hausdorff and maximal compact spaces

d-20 - H-Closed Spaces

On the existence of continuous (approximate) roots of algebraic equations

Let X, Y be compact Hausdorff spaces and B£(X), B*(Y)f 0 < a, |3 < 12 (the first uncountable ordinal), the associated Banach spaces of bounded real-valued Baire functions of classes a and 0. If B*(X) =£ B*AX) (which is the case if a =£ 0 and X is not dispersed), then B*(X) is neither linearly isometric to BQ(Y) nor equivalent to B^(Y) in several other ways. B^(X) is linearly isometric to B%(Y) if and only if X is Baire isomorphic to Y. For 1 < a < ft the maximal ideal space of B£(X) for a nondispersed compact space X As not an F-space. 1. Let I bea compact (more generally, completely regular) Hausdorff space and C(X) the space of continuous real-valued functions on X. Let B0(X) = C(X)9 and inductively define Ba(X) for each ordinal a < Q, (Q denotes the first uncountable ordinal) to be the space of pointwise limits of sequences of functions in U | < a B^(X). Let B*(X) be the space of bounded functions contained in Ba(X). With the pointwise operations Ba(X) and B*(X) are lattice-ordered algebras. With the supremum norm B*(X) is a Banach algebra (see [4, §41]). The Baire sets of X of multiplicative class a, denoted by Za(X)9 are defined to be the zero sets of functions in B*(X). Those of additive class a, denoted by CZa(X), are defined as the complements of sets in Za(X). Finally, those of ambiguous class a, denoted by Aa(X), are the sets which are simultaneously in Za(X) and CZa(X). With the set-theoretic operations of union and intersection, Aa(X) is a Boolean algebra for each a < 12. The sets of exactly ambiguous class a, denoted by EAa(X are those in ^ a ( Z ) U ^ < a Ac(X). The sets of exactly additive and exactly multiplicative class a are defined analogously. The class of all Baire subsets of X is AMS (MOS) subject classifications (1970). Primary 06A65, 26A21, 28A05, 46E1S, 46E30, 46J10, 54C50, 54H05; Secondary 04A15.

Function-space compactifications of function spaces

A categorical characterization of a subcategory S of TOP (or 7J) is one which enables the identification of S in TOP (or 7^) without requiring the reconstruction of the topological structure of its objects.In this paper we so characterize various familiar subcategories of TOP (Hausdorff spaces, normal spaces, compact Hausdorff spaces, paracompact Hausdorff spaces, metrizable spaces, first countable spaces) in terms of the global behavior of the (objects and) morphisms of the subcategory. Introduction. A categorical characterization of a subcategory S of TOP (or of T2) is one which enables the identification of S in TOP (or T2) without requiring the reconstruction of the topological structure of its objects.The proof of the triviality of the automorphism class group of TOP [Fe, p. 32] assures us of the existence of such characterizations.As will be shown in §1, this proof establishes that all the topological structure of an object of TOP can be recovered from the behavior of the arrows with that object as source and/ or target.In §2 we establish a number of lemmas (mostly known), and we use these lemmas in §3 to characterize many important subcategories of TOP and/ or T2, such as the metrizable spaces, the paracompact Hausdorff spaces, the normal spaces, etc.We can distinguish at least three classes of "categorical" characterizations of subcategories of TOP, which are listed below: Class 3. Straightforward translations of the familiar definitions.These can always be obtained because of the results in §1.Example: Characterize the concepts "open", "cover", "refinement", "locally finite" categorically and then the standard definition of paracompact spaces is categorical.Class 2. Characterizations using standard categorical constructions (product, coproduct, epimorphic images, etc.) but in terms of individual objects, and/or constants outside the subcategory.Example: The category of paracompact Hausdorff spaces contains precisely those Hausdorff spaces X such that

By the Riesz Representation Theorem for locally compact Hausdorff spaces, for every positive linear functional I on K(X) there is a measure @m such that I(f)=@!fd@m, where K(X) is the set of continuous real functions with compact support on the locally compact Hausdorff space X. In this article we prove a uniformly computable version of this theorem for computably locally compact computable Hausdorff spaces X. We introduce a representation of the positive linear functionals I on K(X) and a representation of the Borel measures on X and prove that for every such functional I a measure @m can be computed and vice versa such that I(f)=@!fd@m.

Let G be a locally compact second countable Hausdorff space with a positive regular Borel measure λ, where λ is invariant under a continuous injective mapping φ : G → G. We characterize the disjoint hypercyclicity of finite weighted translations generated by φ acting on the weighted space Lp (G, ω) (1 ≤ p &lt; ∞).

On classes of compactifications which are always singular

Let X be a compact Hausdorff space and C(X) the Banach space of all real-valued continuous functions on X under the sup norm. If H is a linear subspace of C(X) which separates the points of X and contains the constant functions, then it is well known that there exists a smallest closed subset of X, called the Silov boundary of X relative to H and denoted by OHX, with the property that each hCH attains its maximum value on afHX. A point xeX is called an H-extremal point if the only positive linear functional u on C(X) such that u(h) = h(x) for all hCH is the evaluation functional 4. where O.,(f) =f(x) for all fE C(X). The set of H-extremal points of X is called the Choquet boundary of X relative to H and will be denoted by VHX. H. Bauer has shown [1, 2.1 ] that the Choquet boundary of X relative to H is nonempty and the Silov boundary of X relative to H is equal to the closure of the Choquet boundary. Bauer has also introduced (see [1 ]) an abstract Dirichlet problem for the above setting. If S:DHX is closed and xCX, then MS denotes the set of positive linear functionals u on C(S) such that u(h I S) =h(x) for all hCH. MS is always nonempty. The measures in MX'HX are called the H-harmonic measures belonging to x. A function f in C(X) is said to be H-harmonic if for every xEX and every u M we have u(f) =f(x). The set of H-harmonic functions is denoted by H. The Dirichlet problem is said to be solvable for X relative to H if 1^ j aHX = c(aHX). The Dirichlet problem is solvable if and only if to each x in X belongs exactly one H-harmonic measure (see [1, Satz 9]). Let X1, X2 be compact Hausdorff spaces and H1, H2 separating linear subspaces of C(X1), C(X2) respectively, which contain the constant functions. Equipped with the usual product topology, the cartesian product X1 X X2 is a compact Hausdorff space. In this paper we show that a subfamily H1+H2 of C(XIXX2) can be defined, in a natural way, for which VH1+H2Xl X X2 exists and VH1+H2Xl X X2 = VH1X1X VH2X2. We then give two theorems on the relation of the

Every closed subset of a compact topological space is compact. Also every compact subset of a Hausdorff topological space is closed. It follows that compact subsets are precisely the closed subsets in a compact Hausdorff space. It is also proved that a topological space is maximal compact if and only if its compact subsets are precisely the closed subsets. A locale is a categorical extension of topological spaces and a frame is an object in its opposite category. We investigate to find whether the closed sublocales are exactly the compact sublocales of a compact Hausdorff frame. We also try to investigate whether the closed sublocales are exactly the compact sublocales of a maximal compact frame.

Supercompact minus compact is super

Compact spaces of diversity two

In this paper we characterize maximal C-compact spaces, maximal QHC spaces, and maximal nearly compact spaces.We also discuss a covering property which turns out to be equivalent to 5-closed and extremally disconnected.1. (-compact spaces.In 1969 Giovanni Viglino [VI] introduced the concept of C-compact spaces and showed that a Hausdorff C-compact space is minimal Hausdorff (and thus maximal C-compact).In this first section we shall characterize all maximal C-compact spaces.The author wishes to thank the referee whose suggestions improved the exposition of this paper.The concept of a C-compact space utilizes the concept of QHC-subspaces.A topological space is quasi-//-closed (QHC) if every open cover has a finite proximate subcover; that is, every open cover has a finite subfamily whose union is dense in the space or equivalently a finite subfamily whose closures cover the space.A subset A of a topological space (X, t) is QHC if it is QHC in its relative topology.A subset A of a topological space (A*, t) is QHC relative to t if every r-open cover of A has a finite subfamily whose T-closures cover the set.A QHC subset is QHC relative to t but not conversely [PT1].A space is C-compact if every closed subset is QHC relative to t.A topological space (A*, t) with property R is maximal R [minimal R] if whenever t* d t [t* c t], (X, t*) does not have property R.The following was proved by Viglino [VI]: In Hausdorff spaces, compact -* C-compact - minimal T2, and the implications are not reversible.From its very definition it is obvious that every C-compact space is QHC.Maximal QHC spaces have been studied by the author in [C2] and included in a general class of spaces in [C3].C-compactness is not of the latter class since the properties included in this class are contractive, contagious, preserved by finite unions, semiregular, and regular closed hereditary.C-compactness is contractive [SI] and preserved by finite unions (Theorem 1), but is not contagious (Example 1), semiregular (Example 2), or regular closed hereditary [SI].A property is contagious if whenever it is possessed by a dense subset of a space then the entire space has the property.Examples of contagious properties are connectedness, separability, and QHC.

Linear extension operators of bounded norms

For a topological space $X$, let $X_\delta$ be the space $X$ with $G_\delta$-topology of $X$. For an uncountable cardinal $\kappa$, we prove that the following are equivalent: (1) $\kappa$ is $\omega_1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindelof degree of $X_\delta$ is $\le \kappa$. (3) For every compact Hausdorff space $X$, the weak Lindelof degree of $X_\delta$ is $\le \kappa$. This shows that the least $\omega_1$-strongly compact cardinal is the supremum of the Lindelof and the weak Lindelof degrees of compact Hausdorff spaces with $G_\delta$-topology. We also prove the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with $G_\delta$-topology. For the square of a Lindelof space, using weak $G_\delta$-topology, we prove that the following are consistent: (1) the least $\omega_1$-strongly compact cardinal is the supremum of the (weak) Lindelof degrees of the squares of regular $T_1$ Lindelof spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular $T_1$ Lindelof spaces.