Productive Coreflective Classes in Top

If sequential cardinals do not exist then every topological space is generated from a converging sequence by using finite products, disjoint sums and quotients. The author acknowledges the support of the grants MSM 0021620839 and GAČR 201/06/0933 of Czech Republic.     

When is the Isbell topology a group topology?

Conditions on a topological space X under which the space C(X,R) of continuous real-valued maps with the Isbell topology κ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in Cκ(X,R) if and only if X is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in Cκ(X,R) if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if X is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.     

Function spaces and one point extensions for the construct of metered spaces

The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M.     

Almost disjoint large subsets of semigroups

There are several notions of largeness in a semigroup S that originated in topological dynamics. Among these are thick, central, syndetic and piecewise syndetic. Of these, central sets are especially interesting because they are partition regular and are guaranteed to contain substantial combinatorial structure. It is known that in (N,+) any central set may be partitioned into infinitely many pairwise disjoint central sets. We extend this result to a large class of semigroups (including (N,+)) by showing that if S is a semigroup in this class which has cardinality κ then any central set can be partitioned into κ many pairwise disjoint central sets. We also show that for this same class of semigroups, if there exists a collection of μ almost disjoint subsets of any member S, then any central subset of S contains a collection of μ almost disjoint central sets. The same statement applies if “central” is replaced by “thick”; and in the case that the semigroup is left cancellative, “central” may be replaced by “piecewise syndetic”. The situation with respect to syndetic sets is much more restrictive. For example, there does not exist an uncountable collection of almost disjoint syndetic subsets of N. We investigate the extent to which syndetic sets can be split into disjoint syndetic sets.     

Thin-type dense sets and related properties

We build on Gruenhage, Natkaniec, and Piotrowski?s study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that X² has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (GC) but not witnessed by a collection of finite sets. We show that the question of when the topological sum of two countable strongly irresolvable spaces satisfies (NC) is related to the Rudin-Keisler order on βω. We also introduce and study the concepts of <κ-thin and superslim dense sets.     

Characters of ultrafilters and tightness of products of fans

As applications of productivity of coreflective classes of topological spaces, the following results will be proved: (1) Characters of points of βN?N are not smaller than any submeasurable cardinal less or equal to ω². (2) If κ is a submeasurable cardinal and S is a sequential fan with κ many spines then the tightness of the κ-power of S is equal to κ. In fact, a little more general results are proved.     

Hereditarily strongly cwH and other separation axioms

We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or full-cardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD(ℵ₁), fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are α-normal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily β-normal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be α-normal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be β-normal; (3) hereditarily cwH spaces which fail to be α-normal. We characterize those regular spaces X such that X×(ω+1) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, non-normal space. The ZFC-independence of several statements involving the hereditarily wD(ℵ₁) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b=ω₁.     

On the Levi and Lebesgue properties

The paper is devoted to a question if the Levi property is preserved by direct sums and quotients. The three-space problem for the Levi and Lebesgue properties in topological Riesz spaces is also investigated.     

On resolvable primal spaces

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If f is a selfmap on X, the sets A?X with f (A)?A are the closed sets of an Alexandroff topology called the primal topology 𝒫(f ) associated with f. We investigate resolvability for primal spaces (X, 𝒫(f)). Our main result is that an Alexandroff space is resolvable if and only if it has no isolated points. Moreover, n-resolvability and other related concepts are investigated for primal spaces.     

“Boundedly generated topological spaces”

The new class of Boundedly generated topological spaces (or l-spaces) is defined and studied by topological methods. It is shown that it is strictly broader than the class of (Hausdorff) compactly generated spaces (or k-spaces) and also that l-spaces possess many of the nice properties of k-spaces e.g. they are closed under the formation of disjoint unions, quotients, direct limits e.t.c. The topology of uniform convergence on boundeda is also studied and in general, it is shown to be strictly finer than the compact-open topology on the space of continuous functions.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

Essentially disjoint families, conflict free colorings and Shelah’s revised GCH

Using Shelah’s revised GCH theorem we prove that if μ<? _ω ≦λ are cardinals, then every μ-almost disjoint family ${\mathcal{A}}\subset {[\lambda]}^{\beth_{\omega}}$ is essentially disjoint, i.e. for each ${A\in {\mathcal{A}}}$ there is a set F(A)∈[A]^<|A| such that the family $\{{A\setminus F(A)}: {A\in {\mathcal{A}}}\}$ is disjoint. We also show that if μ≦κ≦λ are cardinals, κ≧ω, and

• every μ-almost disjoint family ${\mathcal{A}}\subset {[\lambda]}^{{\kappa}}$ is essentially disjoint,

then

• every μ-almost disjoint family ${\mathcal {B}}\subset {[\lambda]}^{\geqq {\kappa}}$ has a conflict-free coloring with κ colors, i.e. there is a coloring f:λ→κ such that for all ${B\in {\mathcal{B}}}$ there is a color ξ<κ such that |{β∈B:f(β)=ξ}|=1.

Putting together these results we obtain that if μ<? _ω ≦λ, then every μ-almost disjoint family ${{\mathcal{B}}\subset {[\lambda]}^{\geqq \beth_{\omega}}}$ has a conflict-free coloring with ? _ω colors. To yield the above mentioned results we also need to prove a certain compactness theorem concerning singular cardinals.     

Infinite distributive laws versus local connectedness and compactness properties

Various local connectedness and compactness properties of topological spaces are characterized by higher degrees of distributivity for their lattices of open (or closed) sets, and conversely. For example, those topological spaces for which not only the lattice of open sets but also that of closed sets is a frame, are described by the existence of web neighborhood bases, where webs are certain specific path-connected sets. Such spaces are called web spaces. The even better linked wide web spaces are characterized by F-distributivity of their topologies, and the worldwide web spaces (or C-spaces) by complete distributivity of their topologies. Similarly, strongly locally connected spaces and locally hypercompact spaces are characterized by suitable infinite distributive laws. The web space concepts are also viewed as natural extensions of spaces that are semilattices with respect to the specialization order and have continuous (unary, binary or infinitary) semilattice operations.     

Polar spaces having some line of cardinality two

The nonthick geometries of type C_n and D_n or equivalently all polar spaces having at least one line of cardinality 2 are classified. It turns out that there are two classes of such polar spaces. On the one hand, decomposable polar spaces or polar spaces which are direct sums of two or more polar spaces are obtained. On the other hand, polar spaces arising from the interval lattice of an irreducible projective geometry which can also be seen as being partitioned by a pair of disjoint maximal singular subspaces can be gotten.     

On the characterization of radical and semisimple classes in categories

We define and characterize radical and semisimple classes in a category K which satisfies certain conditions. These conditions are such that K could be any of the categories of associative rings, groupsR-modules, topological spaces or graphs. Among others, the following is proved:.

A class of objects R in K is a radical class if and only if K is a cohereditary component class which is closed under extensions and with T ? R. A class of objects S in K is a semisimple class if and only if S is a hereditary class which is closed under subdirect embed-dings and extensions with T ? S.     

Countably compact hyperspaces and Frolík sums

Let H⁰(X) (H(X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H(X) is to characterize those X for which H(X) is countably compact. We conjecture that u-compactness of X for some u∈ω^∗ (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction.We define the property R(κ): for every family of closed subsets of X separated by pairwise disjoint open sets and any family of natural numbers, the product is countably compact, and prove that if H(X) is countably compact for a T₂-space X then X satisfies R(κ) for all κ. A space has R(1) iff all its finite powers are countably compact, so this generalizes a theorem of J. Ginsburg: if X is T₂ and H(X) is countably compact, then so is Xn for all n<ω. We also prove that, for κ<t, if the T₃ space X satisfies a weak form of R(κ), the orbit of every point in X is dense, and X contains κ pairwise disjoint open sets, then Xκ is countably compact. This generalizes the following theorem of J. Cao, T. Nogura, and A. Tomita: if X is T₃, homogeneous, and H(X) is countably compact, then so is Xω.Then we study the Frolík sum (also called “one-point countable-compactification”) of a family . We use the Frolík sum to produce countably compact spaces with additional properties (like first countability) whose hyperspaces are not countably compact. We also prove that any product _∏α<κH⁰(Xα) embeds into .     

Uniformly normal structure and uniformly Lipschitzian semigroups

Assume that X is a real Banach space with uniformly normal structure and C is a nonempty closed convex subset of X. We show that a κ-uniformly Lipschitzian semigroup of nonlinear self-mappings of C admits a common fixed point if the semigroup has a bounded orbit and if κ is appropriately greater than one. This result applies, in particular, to the framework of uniformly convex Banach spaces.     

Quotients of uniform spaces

This paper develops the basic theory of quotients of uniform spaces via sufficiently nice group actions. We generalize and unify two fundamental constructions: quotients of topological groups via closed normal subgroups and quotients of metric spaces via actions by isometries. Basic results about inverse limits of topological groups are extended to inverse limits of group actions on uniform spaces, and notions of prodiscrete action and generalized covering map are introduced.     

Indestructibility, instances of strong compactness, and level by level inequivalence

Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ ⁺ strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which ${A = \emptyset}$ . The first of these contains a supercompact cardinal κ and is such that no cardinal δ > κ is measurable, κ’s supercompactness is indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing, and every measurable cardinal δ < κ is δ ⁺ strongly compact. The second of these contains a strong cardinal κ and is such that no cardinal δ > κ is measurable, κ’s strongness is indestructible under < κ-strategically closed, (κ ⁺, ∞)-distributive forcing, and level by level inequivalence between strong compactness and supercompactness holds. The model from the first of our forcing constructions is used to show that it is consistent, relative to a supercompact cardinal, for the least cardinal κ which is both strong and has its strongness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing to be the same as the least supercompact cardinal, which has its supercompactness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing. It further follows as a corollary of the first of our forcing constructions that it is possible to build a model containing a supercompact cardinal κ in which no cardinal δ > κ is measurable, κ is indestructibly supercompact, and every measurable cardinal δ < κ which is not a limit of measurable cardinals is δ ⁺ strongly compact.