On Fuzzy L-paracompact Topological Spaces

The aim of this paper is to study fuzzy extensions of some covering properties defined by L. Kalantan as a modification of some kinds of paracompactness-type properties due to A.V.Arhangels'skii and studied later by other authors. In fact, we obtain that: if (X,T) is a topological space and A is a subset of X, then A is Lindelöf in (X,T) if and only if its characteristic map χ_{A} is a Lindelöf subset in (X,ω(T)). If (X,τ) is a fuzzy topological space, then, (X,τ) is fuzzy Lparacompact if and only if (X,ι(τ)) is L-paracompact, i.e. fuzzy L-paracompactness is a good extension of L-paracompactness. Fuzzy L₂-paracompactness is a good extension of L₂- paracompactness. Every fuzzy Hausdorff topological space (in the Srivastava, Lal and Srivastava' or in the Wagner and McLean' sense) which is fuzzy locally compact (in the Kudri and Wagner' sense) is fuzzy L₂-paracompact

The Reverse of the Intermediate Value Theorem in Some Topological Spaces

Any continuous function with values in a Hausdorff topological space has a closed graph and satisfies the property of intermediate value. However, the reverse implications are false, in general. In this article, we treat additional conditions on the function, and its graph for the reverse to be true.

Quasicontinuous functions and the topology of uniform convergence on compacta

Let X be a Hausdorff topological space, Q(X,R) be the space of all quasicontinuous functions on X with values in R and ?UC be the topology of uniform convergence on compacta. If X is hemicompact, then (Q(X,R), ?UC) is metrizable and thus many cardinal invariants, including weight, density and cellularity coincide on (Q(X,R), ?UC). We find further conditions on X under which these cardinal invariants coincide on (Q(X,R), ?UC) as well as characterizations of some cardinal invariants of (Q(X,R), ?UC). It is known that the weight of continuous functions (C(R,R), ?UC) is ?0. We will show that the weight of (Q(R,R), ?UC) is 2c.

Closed subsets of compact-like topological spaces

<p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of<br />countably pracompact topological spaces. We construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. This example provides an affirmative answer to a question posed by Banakh, Dimitrova, and Gutik in [4]. Also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a Hausdorff topological semigroup whose space is weakly H-closed.</p>

SMALL-BOUND ISOMORPHISMS OF FUNCTION SPACES

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ . For $i=1,2$ , let $K_{i}$ be a locally compact (Hausdorff) topological space and let ${\mathcal{H}}_{i}$ be a closed subspace of ${\mathcal{C}}_{0}(K_{i},\mathbb{F})$ such that each point of the Choquet boundary $\operatorname{Ch}_{{\mathcal{H}}_{i}}K_{i}$ of ${\mathcal{H}}_{i}$ is a weak peak point. We show that if there exists an isomorphism $T:{\mathcal{H}}_{1}\rightarrow {\mathcal{H}}_{2}$ with $\left\Vert T\right\Vert \cdot \left\Vert T^{-1}\right\Vert <2$ , then $\operatorname{Ch}_{{\mathcal{H}}_{1}}K_{1}$ is homeomorphic to $\operatorname{Ch}_{{\mathcal{H}}_{2}}K_{2}$ . We then provide a one-sided version of this result. Finally we prove that under the assumption on weak peak points the Choquet boundaries have the same cardinality provided ${\mathcal{H}}_{1}$ is isomorphic to ${\mathcal{H}}_{2}$ .

Lacunary distributional convergence in topological spaces

Most of the summability methods cannot be defined in an arbitrary Hausdorff topological space unless one introduces a linear or a group structure. In the present paper, using distribution functions over the Borel ?-field of the topology and lacunary sequences we define a new type of convergencemethod in an arbitrary Hausdorff topological space and we study some inclusion theorems with respect to the resulting summability method. We also investigate the inclusion relation between lacunary sequence and lacunary refinement of it.

On quasi-orbital space

<p align="left">Let <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">G </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">be a subgroup of the group Homeo(</span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E</span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">) of homeomorphisms </span>of a Hausdorff topological space <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E</span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">. The class of an orbit </span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">O </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">of </span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">G </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">is the union of </span>all orbits having the same closure as <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">O</span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">. We denote by </span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E=</span></em></span></em><span style="font-family: CMEX8; font-size: xx-small;"><span style="font-family: CMEX8; font-size: xx-small;">e</span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">G </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">the space of classes </span>of orbits called quasi-orbit space. A space <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">X </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">is called a quasi-orbital space if </span>it is homeomorphic to <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E=</span></em></span></em><span style="font-family: CMEX8; font-size: xx-small;"><span style="font-family: CMEX8; font-size: xx-small;">e</span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">G </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">where </span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">is a compact Hausdorff space. In this </span>paper, we show that every in nite second countable quasi-compact <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">T</span></em></span></em><span style="font-family: CMR6; font-size: xx-small;"><span style="font-family: CMR6; font-size: xx-small;">0</span></span><span style="font-family: CMR8; font-size: xx-small;">-space </span>is the quotient of a quasi-orbital space.</p><p align="left"> </p><p align="left"> </p>

A Note on Topological Properties of Non-Hausdorff Manifolds

The notion of compatible apparition points is introduced for non-Hausdorff manifolds, and properties of these points are studied. It is well known that the Hausdorff property is independent of the other conditions given in the standard definition of a topological manifold. In much of literature, a topological manifold of dimension is a Hausdorff topological space which has a countable base of open sets and is locally Euclidean of dimension . We begin with the definition of a non-Hausdorff topological manifold.

Compact and extremally disconnected spaces

Viglino defined a Hausdorff topological space to beC-compact if each closed subset of the space is anH-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is anS-set in the sense of Dickman and Krystock. Such spaces are calledC-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterizeC-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown thatC-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class ofC-s-compact spaces is properly contained in the class ofC-compact spaces as well as in the class ofS-closed spaces of Thompson. In general, a compact space need not beC-s-compact. The product of twoC-s-compact spaces need not beC-s-compact.

Open and Proper Maps Characterized by Continuous Setvalued Maps

In the first part of the paper, given a continuous map f from a Hausdorff topological space X onto a Hausdorff topological space Y, we consider the reciprocal map f* from Y into the collection of closed subsets of X, which maps y ∈ Y to . is endowed with the pseudotopological structure of convergence of closed sets. We will use the filter description of this convergence, as defined by Choquet and Gähler [2], [5], which is equivalent to the “topological convergence” of sets as introduced by Frolík and Mrówka [4], [10]. These notions in fact generalize the convergence of sequences of sets defined by Hausdorff [6]. We show that the continuity of f* is equivalent to the openness of f. On f*(Y), the set of fibers of f, we consider the pseudotopological structure induced by the closed convergence on .