A first countable linearly Lindelöf not Lindelöf topological space

A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. It is well known that every Lindelöf space is linearly Lindelöf. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<ℵω.Arhangel?skii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed ℵ₀². Consequently, a first countable linearly Lindelöf space is Lindelöf if ℵω>ℵ₀². They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. This question is supported by the fact that all known linearly Lindelöf not Lindelöf spaces are of character at least ℵω. We answer this question in the negative by constructing a counterexample from MA+ℵω<ℵ₀².A modification of Alster?s Michael space that is first countable is presented.     

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.     

The D-property in unions of scattered spaces

We show in a direct way that a space is D if it is a finite union of subparacompact scattered spaces. This result cannot be extended to countable unions, since it is known that there is a regular space which is a countable union of paracompact scattered spaces and which is not D. Nevertheless, we show that every space which is the union of countably many regular Lindelöf C-scattered spaces has the D-property. Also, we prove that a space is D if it is a locally finite union of regular Lindelöf C-scattered spaces.     

Weak covering properties and selection principles

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster?s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most _ℵ1${\aleph }_1$. It turns out that topological spaces having Alster?s property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alster?s property satisfy a seemingly stronger version of Alster?s property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property.     

A note on D-spaces

We introduce notions of nearly good relations and N-sticky modulo a relation as tools for proving that spaces are D-spaces. As a corollary to general results about such relations, we show that Cp(X) is hereditarily a D-space whenever X is a Lindelöf Σ-space. This answers a question of Matveev, and improves a result of Buzyakova, who proved the same result for X compact.We also prove that if a space X is the union of finitely many D-spaces, and has countable extent, then X is linearly Lindelöf. It follows that if X is in addition countably compact, then X must be compact. We also show that Corson compact spaces are hereditarily D-spaces. These last two results answer recent questions of Arhangel'skii. Finally, we answer a question of van Douwen by showing that a perfectly normal collectionwise-normal non-paracompact space constructed by R. Pol is a D-space.     

Domination by second countable spaces and Lindelöf Σ-property

Given a space M, a family of sets A of a space X is ordered by M if A={AK:K is a compact subset of M} and K⊂L implies AK⊂AL. We study the class M of spaces which have compact covers ordered by a second countable space. We prove that a space Cp(X) belongs to M if and only if it is a Lindelöf Σ-space. Under MA(ω₁), if X is compact and (X×X)\Δ has a compact cover ordered by a Polish space then X is metrizable; here Δ={(x,x):x∈X} is the diagonal of the space X. Besides, if X is a compact space of countable tightness and X²\Δ belongs to M then X is metrizable in ZFC.We also consider the class M^? of spaces X which have a compact cover F ordered by a second countable space with the additional property that, for every compact set P⊂X there exists F∈F with P⊂F. It is a ZFC result that if X is a compact space and (X×X)\Δ belongs to M^? then X is metrizable. We also establish that, under CH, if X is compact and Cp(X) belongs to M^? then X is countable.     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

Applications of almost one-to-one maps

A continuous map of topological spaces X,Y is said to be almost 1-to-1 if the set of the points x∈X such that f⁻¹(f(x))={x} is dense in X; it is said to be light if pointwise preimages are 0-dimensional. In a previous paper we showed that sometimes almost one-to-one light maps of compact and σ-compact spaces must be homeomorphisms or embeddings. In this paper we introduce a similar notion of an almost d-to-1 map and extend the above results to them and other related maps. In a forthcoming paper we use these results and show that if f is a minimal self-mapping of a 2-manifold then point preimages under f are tree-like continua and either M is a union of 2-tori, or M is a union of Klein bottles permuted by f.     

Generic uniqueness of equilibrium points

Let X be a nonempty, convex and compact subset of normed linear space E (respectively, let X be a nonempty, bounded, closed and convex subset of Banach space E and A be a nonempty, convex and compact subset of X) and f:X×X→R be a given function, the uniqueness of equilibrium point for equilibrium problem which is to find x^∗∈X (respectively, x^∗∈A) such that f(x^∗,y)≥0 for all y∈X (respectively, f(x^∗,y)≥0 for all y∈A) is studied with varying f (respectively, with both varying f and varying A). The results show that most of equilibrium problems (in the sense of Baire category) have unique equilibrium point.     

Some more examples of monotonically Lindelöf and not monotonically Lindelöf spaces

A space is monotonically Lindelöf (mL) if one can assign to every open cover U a countable open refinement r(U) (still covering the space) so that r(U) refines r(V) whenever U refines V. Some examples of mL and non-mL spaces are considered. In particular, it is shown that the product of a mL space and the convergent sequence need not be mL, that some L-spaces are mL, and that Cp(X) is mL only for countable X.     

Continuous selections avoiding extreme points

It is shown that if X is a countably paracompact collectionwise normal space, Y is a Banach space and φ:X→Y² is a lower semicontinuous mapping such that φ(x) is Y or a compact convex subset with Cardφ(x)>1 for each x∈X, then φ admits a continuous selection f:X→Y such that f(x) is not an extreme point of φ(x) for each x∈X. This is an affirmative answer to the problem posed by V. Gutev, H. Ohta and K. Yamazaki [V. Gutev, H. Ohta and K. Yamazaki, Selections and sandwich-like properties via semi-continuous Banach-valued functions, J. Math. Soc. Japan 55 (2003) 499-521].     

Classes defined by stars and neighbourhood assignments

We apply and develop an idea of E. van Douwen used to define D-spaces. Given a topological property P, the class P^∗ dual to P (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment there is Y⊂X with Y∈P and . We prove that the classes of compact, countably compact and pseudocompact are self-dual with respect to neighbourhood assignments. It is also established that all spaces dual to hereditarily Lindelöf spaces are Lindelöf. In the second part of this paper we study some non-trivial classes of pseudocompact spaces defined in an analogous way using stars of open covers instead of neighbourhood assignments.     

Monoreflections of Archimedean ?-groups, regular σ-frames and regular Lindelöf frames

We prove that in the category of Archimedean lattice-ordered groups with weak unit there is no homomorphism-closed monoreflection strictly between the strongest essential monoreflection (the so-called “closure under countable composition”) and the strongest monoreflection (the epicompletion). It follows that in the category of regular σ-frames, the only non-trivial monoreflective subcategory that is hereditary with respect to closed quotients consists of the boolean σ-algebras. Also, in the category of regular Lindelöf locales, there is only one non-trivial closed-hereditary epi-coreflection. The proof hinges on an elementary lemma about the kinds of discontinuities that are exhibited by the elements of a composition-closed l-group of real-valued functions on R.     

Embeddings and frames of function spaces

For every Tychonoff space X we denote by Cp(X) the set of all continuous real-valued functions on X with the pointwise convergence topology, i.e., the topology of subspace of RX. A set P is a frame for the space Cp(X) if Cp(X)⊂P⊂RX. We prove that if Cp(X) embeds in a σ-compact space of countable tightness then X is countable. This shows that it is natural to study when Cp(X) has a frame of countable tightness with some compactness-like property. We prove, among other things, that if X is compact and the space Cp(X) has a Lindelöf frame of countable tightness then t(X)?ω. We give some generalizations of this result for the case of frames as well as for embeddings of Cp(X) in arbitrary spaces.     

The product of two ordinals is hereditarily dually discrete

Let μ and ν be two ordinals. If X is a subspace of μ×ν, then X is dually discrete. This gives a positive answer to a question of Alas, Junqueira and Wilson. By this conclusion and a known conclusion we show that a subspace Y of μ×ν has countable spread if and only if the space Y is hereditarily a Lindelöf D-space.     

Complementation and decompositions in some weakly Lindelöf Banach spaces

Let Γ denote an uncountable set. We consider the questions if a Banach space X of the form C(K) of a given class (1) has a complemented copy of c₀(Γ) or (2) for every c₀(Γ)⊆X has a complemented c₀(E) for an uncountable E⊆Γ or (3) has a decomposition X=A⊕B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely Ks such that C(K) is Lindelöf in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms which are independent of ZFC ± CH. If we assume the P-ideal dichotomy, for every c₀(Γ)⊆C(K) there is a complemented c₀(E) for an uncountable E⊆Γ, which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C(K) for K of height ω+1 where every operator is of the form cI+S for c∈R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C(K) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space Cp(K).     

Function spaces over GO-spaces: Part I

We describe the structure of spaces of continuous step functions over GO-spaces. We establish a relation between the Dedekind completion of a GO-space L and properties of the space of continuous functions from L to 2 with finitely many steps. We use the established relation to prove that a countably compact GO-space L has Lindelöf Cp(L) iff the Dedekind remainder of L is Lindelöf and every compact subspace of L is metrizable. Or equivalently, a countably compact GO-space L has Lindelöf Cp(L) iff every compact subspace of L is metrizable and a Gδ-set in L. Other results are obtained.     

Remarks on countability and star covering properties

In this paper, we construct an example of a T₄ feebly Lindelöf space X which is not star Lindelöf under ℵ₀²=ℵ₁², which gives a partial answer to Alas, Junqueira and Wilson (2011) [1, Question 4].     

Approximations of relations by continuous functions

Let X be a Tychonoff space, C(X) be the space of all continuous real-valued functions defined on X and CL(X×R) be the hyperspace of all nonempty closed subsets of X×R. We prove the following result. Let X be a countably paracompact normal space. The following are equivalent: (a) dimX=0; (b) the closure of C(X) in CL(X×R) with the Vietoris topology consists of all F∈CL(X×R) such that F(x)≠∅ for every x∈X and F maps isolated points into singletons; (c) each usco map which maps isolated points into singletons can be approximated by continuous functions in CL(X×R) with the locally finite topology. From the mentioned result we can also obtain the answer to Problem 5.5 in [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] and to Question 5.5 in [R.A. McCoy, Comparison of hyperspace and function space topologies, Quad. Mat. 3 (1998) 243-258] in the realm of normal, countably paracompact, strongly zero-dimensional spaces. Generalizations of some results from [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] are also given.     

Monolithic spaces and D-spaces revisited

We introduce the classes of monotonically monolithic and strongly monotonically monolithic spaces. They turn out to be reasonably large and with some nice categorical properties. We prove, in particular, that any strongly monotonically monolithic countably compact space is metrizable and any monotonically monolithic space is a hereditary D-space. We show that some classes of monolithic spaces which were earlier proved to be contained in the class of D-spaces are monotonically monolithic. In particular, Cp(X) is monotonically monolithic for any Lindelöf Σ-space X. This gives a broader view of the results of Buzyakova and Gruenhage on hereditary D-property in function spaces.