Let $\mathfrak{P}$ be a topological property. We study the relation between the order structure of the set of all $\mathfrak{P}$-extensions of a completely regular space $X$ with compact remainder (partially ordered by the standard partial order $\leq$) and the topology of certain subspaces of the outgrowth $βX\setminus X$. The cases when $\mathfrak{P}$ is either pseudocompactness or realcompactness are studied in more detail.
In this paper we discuss what kind of constrains combinatorial covering properties of Menger, Scheepers, and Hurewicz impose on remainders of topological groups. For instance, we show that such a remainder is Hurewicz if and only it is $σ$-compact. Also, the existence of a Scheepers non-$σ$-compact remainder of a topological group follows from CH and yields a $P$-point, and hence is independent of ZFC. We also make an attempt to prove a dichotomy for the Menger property of remainders of topological groups in the style of Arhangel'skii.
We study a topological generalization of ideal co-maximality in topological rings and present some of its properties, including a generalization of the Chinese remainder theorem. Using the hyperspace uniformity, we prove a stronger version of this theorem concerning infinitely many ideals in supercomplete, pseudo-valuated rings. Finally we prove two interpolation theorems.
When does a topological group $G$ have a Hausdorff compactification $bG$ with a remainder belonging to a given class of spaces? In this paper, we mainly improve some results of A.V. Arhangel'ski\vı and C. Liu's. Let $G$ be a non-locally compact topological group and $bG$ be a compactification of $G$. The following facts are established: (1) If $bG\setminus G$ has a locally a point-countable $p$-metabase and $π$-character of $bG\setminus G$ is countable, then $G$ and $bG$ are separable and metrizable; (2) If $bG\setminus G$ has locally a $δθ$-base, then $G$ and $bG$ are separable and metrizable…
In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in $\mathbf{ZF}$. Among other independence results, the characterization of locally compact Hausdorff spaces having all non-empty metrizable compact spaces as remainders, obtained by Hatzenhuhler and Mattson in $\mathbf{ZFC}$, is proved to be independent of $\mathbf{ZF}$. Urysohn's Metrization Theorem is generalized to the following theorem: every $T_3$-space which admits a base expressible as a …
In this work, we Extend Pawlak approximation spaces by topological spaces. Also, Rough Membership, equality and inclusion relations are extended using topological near open sets. In addition, new extended measures of accuracy and quality of approximations are defined and studied. An application example of data reduction of information systems is introduced.
It is proved that there is no structure of left (right) cancelative semigroup on $[L]$-dimensional universal space for the class of separable compact spaces of extensional dimension $\le [L]$. Besides, we note that the homeomorphism group of $[L]$-dimensional space whose nonempty open sets are universal for the class of separable compact spaces of extensional dimension $\le [L]$ is totally disconnected.
A space $Y$ is called an extension of a space $X$ if $Y$ contains $X$ as a dense subspace. An extension $Y$ of $X$ is called a one-point extension of $X$ if $Y\backslash X$ is a singleton. P. Alexandroff proved that any locally compact non-compact Hausdorff space $X$ has a one-point compact Hausdorff extension, called the one-point compactification of $X$. Motivated by this, S. Mrówka and J.H. Tsai [On local topological properties. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 1035-1040] posed the following more general question: For what pairs of topological propertie…
We study FTOP(L), a fuzzy category with fuzzy functions in the role of morphisms. This category has the same objects as the category L-TOP of Chang-Goguen L-topological spaces,but an essentially wider class of morphisms - so called fuzzy functions introduced earlier in our joint work with U. Hohle and H. Porst.
Let ${\mathscr P}$ be a topological property. We say that a space $X$ is ${\mathscr P}$-connected if there exists no pair $C$ and $D$ of disjoint cozero-sets of $X$ with non-${\mathscr P}$ closure such that the remainder $X\backslash(C\cup D)$ is contained in a cozero-set of $X$ with ${\mathscr P}$ closure. If ${\mathscr P}$ is taken to be "being empty" then ${\mathscr P}$-connectedness coincides with connectedness in its usual sense. We characterize completely regular ${\mathscr P}$-connected spaces, with ${\mathscr P}$ subject to some mild requirements. Then, we study conditions under which …
A twisted sum in the category of topological abelian groups is a short exact sequence $0\to Y\to X \to Z\to 0$ where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to $0\to Y\to Y \times Z \to Z\to 0$. \par We study the class $\ST$ of topological groups $G$ for which every twisted sum $0\to \T\to X \to G\to 0$ splits. We prove that this class contains locally precompact groups, sequential direct limits of locally compact groups and topological groups with $\mathcal{L}_\infty$ topologies. We also prove that it is closed by taking ope…
We study algebraic properties of the Brandt $λ^0$-extensions of monoids with zero and non-trivial homomorphisms between the Brandt $λ^0$-extensions of monoids with zero. We introduce finite, compact topological Brandt $λ^0$-extensions of topological semigroups and countably compact topological Brandt $λ^0$-extensions of topological inverse semigroups in the class of topological inverse semigroups and establish the structure of such extensions and non-trivial continuous homomorphisms between such topological Brandt $λ^0$-extensions of topological monoids with zero. We also describe a category w…
In this paper, we introduce the notion of topologically Banach contraction mapping defined on an arbitrary topological space X with the help of a continuous function $g:X\times X\rightarrow \mathbb{R}$ and investigate the existence of fixed points of such mapping. Moreover, we introduce two types of mappings defined on a non-empty subset of X and produce sufficient conditions which will ensure the existence of best proximity points for these mappings. Our best proximity point results also extend some existing results from metric spaces or Banach spaces to topological spaces. More precisely, ou…
We give a characterization of countable discrete subspace $A$ of a topological space $X$ such that there exists a (linear) continuous mapping $φ:C_p^*(A)\to C_p(X)$ with $φ(y)|_A=y$ for every $y\in C_p^*(A)$. Using this characterization we answer two questions of A.~Arhangel'skii. Moreover, we introduce the notion of well-covered subset of a topological space and prove that for well-covered functionally closed subset $A$ of a topological space $X$ there exists a linear continuous mapping $φ:C_p(A)\to C_p(X)$ with $φ(y)|_A=y$ for every $y\in C_p(A)$.
The article is devoted to microbundles over topological rings. Their structure, homomorphisms, automorphisms and extensions are studied. Moreover, compactifications and inverse spectra of microbundles over topological rings are investigated.
A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties of iso-dense spaces are investigated. A new permutation model is constructed in which a discrete weakly Dedekind-finite space can have the Cantor set as a remainder. A metrization theorem for a class of quasi-metric spaces is deduced. The statement "every compact scattered metrizable space is separable" and several other statements about metric iso-dense spa…
Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object", "everything can possibly exist, unless it yields contradiction", "the ideal elements correctly determine the real things". Here we give a precise formulation of these principles within the framework of the Topological Extensions of [8], structures that generalize at once compactifications, completions, and nonstandard extensions. In this topological context, the above Leibniz's principles appear as a property of separation, a property of compactness, and a prop…
We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its the Stone-Čech compactification $βS$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:S\to Y$ to a second countable space $Y$ can be written as the composition $f=g\circ p$ of an open map $p:X\to Z$ onto a second countable space $Z$ and a map $g:Z\to Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.
On taking a non-trivial and semi-transitive bi-relation constituted by two (hard and soft) binary relations, we report a (i) p-continuity assumption that guarantees the completeness and transitivity of its soft part, and a (ii) characterization of a connected topological space in terms of its attendant properties on the space. Our work generalizes antecedent results in applied mathematics, all following Eilenberg (1941), and now framed in the context of a parametrized-topological space. This re-framing is directly inspired by the continuity assumption in Wold (1943-44) and the mixture-space st…
We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a se…
In the paper we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, $ω$-boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) to\-pological Brandt $λ_i^0$-extensions of semitopological monoids with zero. In particular we show that if $\big\{ \big(B^0_{λ_i}(S_i),τ^0_{B(S_i)}\big) \colon i\in\mathscr{I}\big\}$ is a family of Hausdorff pseudocompact to\-pological Brandt $λ_i^0$-extensions of pseudocompact semitopological monoids with zero such…
We prove several rigidity results for corona $C^*$-algebras and Čech-Stone remainders under the assumption of Forcing Axioms. In particular, we prove that a strong version of Todorčević's $\OCA$ and Martin's Axiom at level $\aleph_1$ imply: (i) that if $X$ and $Y$ are locally compact second countable topological spaces, then all homeomorphisms between $βX\setminus X$ and $βY\setminus Y$ are induced by homeomorphisms between cocompact subspaces of $X$ and $Y$; (ii) that all automorphisms of the corona algebra of a separable $C^*$-algebra are trivial in a topological sense; (iii) that if $A$ is …
In this paper we introduce the concept of MV-topology, a special class of fuzzy topological spaces, and prove a proper extension of Stone Duality to the categories of limit cut complete MV-algebras and Stone MV-spaces, namely, zero-dimensional compact Hausdorff MV-topological spaces. Then we describe the object class of limit cut complete MV-algebras, and show that any semisimple MV-algebra has a limit cut completion, namely, a minimum limit cut complete extension. Last, we compose our duality with other known ones, thus obtaining new categorical equivalences and dualities involving categories…
