Permutable pairs of quasi-uniformities

We continue investigating the lattice (q(X),⊆) of quasi-uniformities on a set X. In particular in this article we start investigating permutable pairs of quasi-uniformities. Among other things, we show that the Pervin quasi-uniformity of a topological space X permutes with its conjugate if and only if X is normal and extremally disconnected.     

The smallest basically disconnected preimage of a space

We show that for each space X, there exists a smallest basically disconnected perfect irreducible preimage ΛX. A corollary of the existence of ΛX is that each locally compact and basically disconnected space X has a smallest basically disconnected compactification.     

Atoms, anti-atoms and complements in the lattice of quasi-uniformities

We describe the atoms of the complete lattice (q(X),⊆) of all quasi-uniformities on a given (nonempty) set X. We also characterize those anti-atoms of (q(X),⊆) that do not belong to the quasi-proximity class of the discrete uniformity on X. After presenting some further results on the adjacency relation in (q(X),⊆), we note that (q(X),⊆) is not complemented for infinite X and show how ideas about resolvability of (bi)topological spaces can be used to construct complements for some elements of (q(X),⊆).     

Stone-Čech remainders of nowhere locally compact spaces

The remainder of a compactification αX of a space X is the space αX - X. For a nowhere locally compact X, it is shown that if αX - X is extremally disconnected, then αX = βX. Conditions on X are obtained which characterize when βX - X is extremally disconnected. Also, conditions are provided which characterize when βX - X is zero-dimensional.     

Infima of quasi-uniform anti-atoms

Let (q(X),⊆) denote the lattice consisting of the set q(X) of all quasi-uniformities on a set X, ordered by set-theoretic inclusion ⊆. We observe that a quasi-uniformity on X is the supremum of atoms of (q(X),⊆) if and only if it is totally bounded and transitive. Each quasi-uniformity on X that is totally bounded or has a linearly ordered base is shown to be the infimum of anti-atoms of (q(X),⊆). Furthermore, each quasi-uniformity U on X such that the topology of the associated supremum uniformity Us is resolvable has the latter property.     

Stone duality and Gleason covers through de Vries duality

We introduce zero-dimensional de Vries algebras and show that the category of zero-dimensional de Vries algebras is dually equivalent to the category of Stone spaces. This shows that Stone duality can be obtained as a particular case of de Vries duality. We also introduce extremally disconnected de Vries algebras and show that the category of extremally disconnected de Vries algebras is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. As a result, we give a simple construction of the Gleason cover of a compact Hausdorff space by means of de Vries duality. We also discuss the insight that Stone duality provides in better understanding of de Vries duality.     

More about complements of quasi-uniformities

We continue our investigations on the lattice (q(X),⊆) of quasi-uniformities on a set X. Improving on earlier results, we show that the Pervin quasi-uniformity (resp. the well-monotone quasi-uniformity) of an infinite topological T₁-space X does not have a complement in (q(X),⊆). We also establish that a hereditarily precompact quasi-uniformity inducing the discrete topology on an infinite set X does not have a complement in (q(X),⊆).     

Scattered compactifications and the orderability of scattered spaces

A space is defined to be suborderable if it is embeddable in a (totally) orderable space. The length of a scattered space X is the least ordinal a such that X(_a), the ath derived set of X, is empty. It is shown that a suborderable scattered space of countable length is hereditarily paracompact, orderable, and admits an orderable scattered compactification.     

Non-regular power homogeneous spaces

We show that the cardinality of any space X with Δ-power homogeneous semiregularization that is either Urysohn or quasiregular is bounded by ²c(X)πχ(X). This improves a result of G.J. Ridderbos who showed this bound holds for Δ-power homogeneous regular spaces. By introducing the notion of a local πθ-base, we show that this bound can be further sharpened. We also show that no H-closed extremally disconnected space is power homogeneous. This is a variation of a result of K. Kunen who showed that no compact F-space is power homogeneous.     

Group action on zero-dimensional spaces

Let X be a Tychonoff space, H(X) the group of all self-homeomorphisms of X and the evaluation function. Call an admissible group topology on H(X) any topological group topology on H(X) that makes the evaluation function a group action. Denote by LH(X) the upper-semilattice of all admissible group topologies on H(X) ordered by the usual inclusion. We show that if X is a product of zero-dimensional spaces each satisfying the property: any two non-empty clopen subspaces are homeomorphic, then LH(X) is a complete lattice. Its minimum coincides with the clopen-open topology and with the topology of uniform convergence determined by a T₂-compactification of X to which every self-homeomorphism of X continuously extends. Besides, since the left, the right and the two-sided uniformities are non-Archimedean, the minimum is also zero-dimensional. As a corollary, if X is a zero-dimensional metrizable space of diversity one, such as for instance the rationals, the irrationals, the Baire spaces, then LH(X) admits as minimum the closed-open topology induced by the Stone-?ech-compactification of X which, in the case, agrees with the Freudenthal compactification of X.     

The notion of o-tightness and C-embedded subspaces of products

We consider the question: when is a dense subset of a space XC-embedded in X? We introduce the notion of o-tightness and prove that if each finite subproduct of a product X = Π_α?AX_α has a countable o-tightness and Y is a subset of X such that π_B(Y) = Π_α?BX_α for every countable B ? A, then Y is C-embedded in X. This result generalizes some of Noble and Ulmer's results on C-embedding.     

Modal Logics of Stone Spaces

Interpreting modal diamond as the closure of a topological space, we axiomatize the modal logic of each metrizable Stone space and of each extremally disconnected Stone space. As a corollary, we obtain that S4.1 is the modal logic of the Pelczynski compactification of the natural numbers and S4.2 is the modal logic of the Gleason cover of the Cantor space. As another corollary, we obtain an axiomatization of the intermediate logic of each metrizable Stone space and of each extremally disconnected Stone space. In particular, we obtain that the intuitionistic logic is the logic of the Pelczynski compactification of the natural numbers and the logic of weak excluded middle is the logic of the Gleason cover of the Cantor space.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

Tightness relative to some (co)reflections in topology

We address what might be termed the reverse reflection problem: given a monoreflection from a category A onto a subcategory B, when is a given object B ∈ B the reflection of a proper subobject? We start with a well known specific instance of this problem, namely the fact that a compact metric space is never the ?ech-Stone compactification of a proper subspace. We show that this holds also in the pointfree setting, i.e., that a compact metrizable locale is never the ?ech-Stone compactification of a proper sublocale. This is a stronger result than the classical one, but not because of an increase in scope; after all, assuming weak choice prin­ciples, every compact regular locale is the topology of a compact Hausdorff space. The increased strength derives from the conclusion, for in general a space has many more sublocales than subspaces. We then extend the analysis from metric locales to the broader class of perfectly normal locales, i.e., those whose frame of open sets consists entirely of cozero elements. We include a second proof of these results which is purely algebraic in character.

At the opposite extreme from these results, we show that an extremally disconnected locale is a compactification of each of its dense sublocales. Finally, we analyze the same phenomena, also in the pointfree setting, for the 0-dimensional compact reflec­tion and for the Lindelöf reflection.     

Submaximal and door compactifications

In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:

(i)

D is co-finite in K(X);

(ii)

for each x∈K(X)?D, {x} is closed.

If X is a noncompact space, then we show that K(X) is a door space if and only if X is a discrete space and K(X) is the one-point compactification of X.     

Topologizing homeomorphism groups of rim-compact spaces

Let X be a Tychonoff space, H(X) the group of all self-homeomorphisms of X with the usual composition and the evaluation function. Topologies on H(X) providing continuity of the evaluation function are called admissible. Topologies on H(X) compatible with the group operations are called group topologies. Whenever X is locally compact T₂, there is the minimum among all admissible group topologies on H(X). That can be described simply as a set-open topology, further agreeing with the compact-open topology if X is also locally connected. We show the same result in two essentially different cases of rim-compactness. The former one, where X is rim-compact T₂ and locally connected. The latter one, where X agrees with the rational number space Q equipped with the euclidean topology. In the first case the minimal admissible group topology on H(X) is the closed-open topology determined by all closed sets with compact boundaries contained in some component of X. Moreover, whenever X is also separable metric, it is Polish. In the rational case the minimal admissible group topology on H(Q) is just the closed-open topology. In both cases the minimal admissible group topology on H(X) is closely linked to the Freudenthal compactification of X. The Freudenthal compactification in rim-compactness plays a key role as the one-point compactification does in local compactness. In the rational case we investigate whether the fine or Whitney topology on H(Q) induces an admissible group topology on H(Q) stronger than the closed-open topology.     

Zero-separating algebras of continuous functions

Let A be a lattice-ordered algebra endowed with a topology compatible with the structure of algebra. We provide internal conditions for A to be isomorphic as lattice-ordered algebras and homeomorphic to Ck(X), the lattice-ordered algebra C(X) of real continuous functions on a completely regular and Hausdorff topological space X, endowed with the topology of uniform convergence on compact sets. As a previous step, we determine this topology among the locally m-convex topologies on C(X) with the property that each order closed interval is bounded.     

On the space of subgroups of a compact group II

Let G be a compact group. Let S(G), C(G), N(G) be the spaces of closed subgroups, cosets of closed subgroups, normal closed subgroups (respectively) of G, with the Vietoris topology.Then: (1) S(G) and C(G) are never connected; (2) N(G) is always totally disconnected; (3) C(G) is totally disconnected if and only if G is totally disconnected; and (4) S(G) is totally disconnected if and only if G/Z(G) is totally disconnected.Further: for totally disconnected G (equivalently, profinite G) (5) S(G), C(G) and N(G) are κ-metrisable; (6) S(G), C(G) and N(G) are Dugundji compact if G has small weight; and (7) consequences for field extensions are derived.     

Note on separate continuity and the Namioka property

A pair 〈B,K〉 is a Namioka pair if K is compact and for any separately continuous , there is a dense A⊆B such that f is ( jointly) continuous on A×K. We give an example of a Choquet space B and separately continuous such that the restriction fΔ_| to the diagonal does not have a dense set of continuity points. However, for K a compact fragmentable space we have: For any separately continuous and for any Baire subspace F of T×K, the set of points of continuity of is dense in F. We say that 〈B,K〉 is a weak-Namioka pair if K is compact and for any separately continuous and a closed subset F projecting irreducibly onto B, the set of points of continuity of fF_| is dense in F. We show that T is a Baire space if the pair 〈T,K〉 is a weak-Namioka pair for every compact K. Under (CH) there is an example of a space B such that 〈B,K〉 is a Namioka pair for every compact K but there is a countably compact C and a separately continuous which has no dense set of continuity points; in fact, f does not even have the Baire property.