A solution to Dilworth's congruence lattice problem

We construct an algebraic distributive lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R.P. Dilworth, from the forties. The lattice D has a compact top element and ℵω+1 compact elements. Our results extend to any algebra possessing a congruence-compatible structure of a join-semilattice with a largest element.     

Preorderable topologies and order-representability of topological spaces

We furnish characterizations of topologies that coincide with the lower topology or with the order topology of some total preorder defined on a set. Leaning on these characterizations we introduce some applications to the study of continuous and semicontinuous order-representability properties of topological spaces.     

Order representability in groups and vector spaces

In the present paper, first we study in a systematic way the numerical representation problem for total preorders defined either on groups or on real vector spaces. Then, we consider groups and real vector spaces equipped with a topology, and analyze the fulfillment of the so-called continuous representability property; the latter meaning that every continuous total preorder defined on the given topological space admits a continuous real-valued order-preserving function. We also explore the analogous cases as above for total preorders that are compatible with the given algebraic structure, looking for real-valued, continuous or not, order-preserving functions that, in addition, are algebraic homomorphisms.     

Representable topologies and locally connected spaces

The objective of this paper is to study certain order-theoretic properties of locally connected topologies. In the main theorem we prove that the assertion that a locally connected topology which satisfies the countable chain condition has the continuous representability property is undecidable in ZFC set theory. Some related problems and generalizations are also included.     

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.     

The convexity lattice of a poset

The authors investigate the lattice Co(P) of convex subsets of a general partially ordered set P. In particular, they determine the conditions under which Co(P) and Co(Q) are isomorphic; and give necessary and sufficient conditions on a lattice L so that L is isomorphic to Co(P) for some P.     

H-spectral spaces

Let X be a T₀-space, we say that X is H-spectral if its T₀-compactification is spectral. This paper deal with topological properties of H-spectral spaces. In the case of T₁-spaces the T₀-compactification coincides with the Wallman compactification. We give necessary and sufficient condition on the T₁-space X in order to get its Wallman compactification spectral.     

Brush spaces and the fixed point property

We introduce the notions of a brush space and a weak brush space. Each of these spaces has a compact connected core with attached connected fibers and may be either compact or non-compact. Many spaces, both in the Hausdorff non-metrizable setting and in the metric setting, have realizations as (weak) brush spaces. We show that these spaces have the fixed point property if and only if subspaces with core and finitely many fibers have the fixed point property. This result generalizes the fixed point result for generalized Alexandroff/Urysohn Squares in Hagopian and Marsh (2010) [4]. We also look at some familiar examples, with and without the fixed point property, from Bing (1969) [1], Connell (1959) [3], Knill (1967) [7] and note the brush space structures related to these examples.     

The structure of Valdivia compact lines

We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The conjecture asserted that a compact line is Valdivia compact if its weight does not exceed ℵ₁, every point of uncountable character is isolated from one side and every closed first countable subspace is metrizable. It turns out that the last condition is not sufficient. On the other hand, we show that the conjecture is valid if the closure of the set of points of uncountable character is scattered. This improves an earlier result of the first author.     

A strong characterization on the ΩEP-property

In this paper a result of A. Illanes and J.J. Charatonik obtained in [J.J. Charatonik, A. Illanes, Mappings on dendrites, Topology Appl. 144 (2004) 109-132, Corollary 5.14] is extended, by showing that a locally connected continuum X has the nonwandering-eventually-periodic property. (ΩEP-property) iff X is a dendrite that does not contain a homeomorphic copy of the null-comb. Also using “An engine breaking the ΩEP-property” constructed by P. Pyrih et al. in [P. Pyrih, J. Hladký, J. Novák, M. Sterzik, M. Tancer, An engine breaking the ΩEP-property, Topology Appl. 153 (2006) 3621-3626] the results obtained in [J.J. Charatonik, A. Illanes, Mappings on dendrites, Topology Appl. 144 (2004) 109-132; H. Méndez-Lango, On the ΩEP-property, Topology Appl. 154 (2007) 2561-2568] and [P. Pyrih, J. Hladký, J. Novák, M. Sterzik, M. Tancer, An engine breaking the ΩEP-property, Topology Appl. 153 (2006) 3621-3626] are extended, by proving that every nonlocally connected continuum X that contains a nondegenerate arc A and a point p∈A such that X is not connected in kleinen at p does not have the ΩEP-property. Answering Question 1 of [P. Pyrih, J. Hladký, J. Novák, M. Sterzik, M. Tancer, An engine breaking the ΩEP-property, Topology Appl. 153 (2006) 3621-3626]. Finally an uncountable family of non-locally connected continua containing arcs with the ΩEP-property is shown.     

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.     

Countable connected Hausdorff and Urysohn bunches of arcs in the plane

In this paper, we answer a question by Krasinkiewicz, Reńska and Sobolewski by constructing countable connected Hausdorff and Urysohn spaces as quotient spaces of bunches of arcs in the plane. We also consider a generalization of graphs by allowing vertices to be continua and replacing edges by not necessarily connected sets. We require only that two “vertices” be in the same quasi-component of the “edge” that contains them. We observe that if a graph G cannot be embedded in the plane, then any generalized graph modeled on G is not embeddable in the plane. As a corollary we obtain not planar bunches of arcs with their natural quotients Hausdorff or Urysohn. This answers another question by Krasinkiewicz, Reńska and Sobolewski.     

Metrizable fiberings of continuous images of the lexicographic square

A Tychonoff space A is metrizably fibered if and only if there exists a continuous map onto a metrizable space B such that for each b∈B,F⁻¹(b) is metrizable. We resolve a question stated by V. Tkachuk by showing that every first countable Hausdorff continuous image of the lexicographic square is metrizably fibered. We also observe that an example of S. Mardeši? and P. Papi? resolves a related question stated by Tkachuk.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.     

Lifting paths on quotient spaces

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn-Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g∈G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.     

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G.We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.     

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.     

Finitely generated varieties of distributive double <Emphasis Type="Italic">p</Emphasis>-algebras universal modulo a group

For any monoid M, any universal variety contains arbitrarily large algebras whose endomorphism monoid is isomorphic to M. A variety universal modulo a group G contains arbitrarily large algebras whose endomorphism monoid is isomorphic to the direct product M x G. One of the results of this paper structurally characterizes all finitely generated varieties of distributive double p-algebras universal modulo a group, and shows that any unavoidable direct factor G is a Boolean group with at most eight elements.