In this article, we consider $\mathcal{I}^\mathcal{K}$-convergence to define a new concept of convergence namely, $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence which generalizes the notion of $\mathcal{S}$-$\mathcal{I}$-convergence introduced by Guevara et al. \cite{GSR20} recently. Some properties of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence of sequences and its relation with compact sets are discussed. In particular, we investigate the relation between semi-compactness and semi-Lindeloffness by introducing the notion of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster point of a sequence. The "Equivalence between semi-dense and dense sets" is utilized to characterize the set of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster points of a sequence as semi-closed subsets of a topological space. Moreover, in product space, we obtain some results for $\mathcal{I}^\mathcal{K}$-convergence which also holds for $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence.
In this paper we have studied some important topological properties and characterization of I^K-convergence of functions which is a common generalization of I*-convergence of functions. We also introduce the idea of I^K*-convergence and I^K-limit points of functions.
The natural duality between "topological" and "regular," both considered as convergence space properties, extends naturally to p-regular convergence spaces, resulting in the new concept of a p-topological convergence space. Taking advantage of this duality, the behavior of p-topological and p-regular convergence spaces is explored, with particular emphasis on the former, since they have not been previously studied. Their study leads to the new notion of a neighborhood operator for filters, which in turn leads to an especially simple characterization of a topology in terms of convergence criter…
In this paper we introduce a weak version of level and epigraph convergence for level functions on topological spaces. In the particular case of topological groups we are able to define convolutions in the set of level functions and show that any such function is the limit in level and epigraph of robust functions.
We show that it is consistent to have an uncountable sequential group of intermediate sequential order while no countable such groups exist. This is proved by adding $ω_2$ Cohen reals to a model of $\diamondsuit$.
In this paper we have shown that a double sequence in a topological space satisfies certain conditions which in turn are capable to generate a topology on a non empty set. Also we have used the idea of I-convergence of double sequences to study the idea of I-sequentially compactness [3] in the sense of double sequences.
In this paper we have used the idea of I-convergence of sequences and nets to study certain conditions of convergence in a topological space. It has been shown separately that a class of sequences and a class of nets in a non-empty set X which are respectively called I-convergence class of sequences and I-convergence class of nets satisfying these conditions generate a topology on X. Further we have correlated the classes of I-convergent sequences and nets with respect to these topologies with the given classes which satisfy these conditions.
We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter $D$, the notions of $D$-compactness and of $D$-pseudocompactness are equivalent. Any product of initially $λ$-compact generalized ordered topological spaces is still initially $λ$-compact. On the other hand, preservation under products of certain compactness properties are independent from the usual axioms for set theory.
In this paper we detect topological Clifford semigroups which are embeddable into Tychonoff products of topological semilattices and cones over topological groups. Also we detect topological Clifford semigroups which embed into compact topological Clifford semigroups.
In this paper we investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterise graph inverse semigroups which admit a compact semigroup topology and describe graph inverse semigroups which can be embeded densely into CLP-compact topological semigroups.
In this paper, we introduce the notion of $\mathcal{M}$-convergence and $\mathcal{MN}$-convergence structures in posets, which, in some sense, generalise the well-known Scott-convergence and order-convergence structures. As results, we give a necessary and sufficient conditions for each generalised convergence structures being topological. These results then imply the following two well-established results: (1) The Scott-convergence structure in a poset $P$ is topological if and only if $P$ is continuous, and (2) The order-convergence structure in a poset $P$ is topological if and only if $P$ …
In this paper, we obtain some results on the relationships between different ideal \linebreak convergence modes namely, $\mathcal{I}^\mathcal{K}$, $\mathcal{I}^{\mathcal{K}^*}$, $\mathcal{I}$, $\mathcal{K}$, $\mathcal{I} \cup \mathcal{K}$ and $(\mathcal{I} \cup \mathcal{K})^*$. We introduce a topological space namely $\mathcal{I}^\mathcal{K}$-sequential space and show that the class of $\mathcal{I}^\mathcal{K}$-sequential spaces contain the sequential spaces. Further $\mathcal{I}^\mathcal{K}$-notions of cluster points and limit points of a function are also introduced here. For a given sequenc…
A new class of functions called L-fuzzy weakly Semi-Preopen (Semi-Preclosed) functions in L-fuzzy topological spaces are introduced in this paper. Some characterizations of this class and its properties and the relationship with other classes of functions between L-fuzzy topological spaces are also obtained.
This paper introduces semiopen and semiclosed soft sets in soft topological spaces. The notions of interior and closure are generalized using these sets. A detail study is carried out on properties of semiopen, semiclosed soft sets, semi interior and semi closure of a soft set in a soft topological space. Various forms of soft functions, like semicontinuous, irresolute, semiopen soft functions are introduced and characterized. Further soft semicompactness,soft semiconnectedness and soft semiseparation axioms are introduced and studied.
We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice $L$ with a monotonic map $\lim_L$ from the lattice of filters on $L$ to $L$, meant to be an abstract version of the map sending every filter of subsets to its set of limits. This construction exhibits the category of convergence spaces as a coreflective subcategory of the opposite of the category of convergence lattices. We extend this construction to coreflections between limit spaces and the opposite of so-called limit lattices and limit coframes, between pretopological convergence spa…
In this paper we characterise graph inverse semigroups which admit only discrete locally compact semigroup topology. This characterization provides a complete answer on the question of Z. Mesyan, J. D. Mitchell, M. Morayne and Y. H. Péresse posed in their paper "Topological graph inverse semigroups".
In this paper, we consider certain topological properties along with certain types of mappings on these spaces defined by the notion of ideal convergence. In order to do that, we primarily follow in the footsteps of the earlier studies of ideal convergence done by using functions (from an infinite set $S$ to $X$) in \cite{CS, das4, das5}, as that is the most general perspective and use functions instead of sequences/nets/double sequences etc. This functional approach automatically provides the most general settings for such studies and consequently extends and unifies the proofs of several old…
In this paper, we continue studying the properties of $γ^{*}$-semi-open sets in topological spaces introduced by S. Hussain, B. Ahmad and T. Noiri[8]. We also introduce and discuss the $γ^{*}$-semi-continuous functions which generalize semi-continuous functions defined by N. Levine [10].
In the paper we describe the structure of $\mathscr{AH}$-completions and $\mathscr{H}$-completions of the discrete semilattices $(\mathbb{N},\min)$ and $(\mathbb{N},\max)$. We give an example of an $\mathscr{H}$-complete topological semilattice which is not $\mathscr{AH}$-complete. Also we construct an $\mathscr{H}$-complete topological semilattice of cardinality $λ$ which has $2^λ$ many open-and-closed continuous homomorphic images which are not $\mathscr{H}$-complete topological semilattices. The constructed examples give a negative answer to Question 17 from the paper J. W. Stepp, {\it Alge…
We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $λ_s$ (the algebraic convergence) and $λ_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov cube respectively. In particular we show that $λ_{ls}$ is a topological convergence iff forcing by B does not produce new reals and that $λ_{ls}$ is weakly topological if B satisfies condition $(\hbar)$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $λ_{ls}$ is a weakly topological convergence, then B is a $2^{\mathfrak h}$-cc algebra or in some generic extensi…
We study the structure of inverse primitive feebly compact semitopological and topological semigroups. We find conditions when the maximal subgroup of an inverse primitive feebly compact semitopological semigroup $S$ is a closed subset of $S$ and describe the topological structure of such semiregular semitopological semigroups. Later we describe the structure of feebly compact topological Brandt $λ^0$-extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In particular we show that inversion in a quasi-regular primitive inverse feebly com…
Let $T\times X\rightarrow X, (t,x)\mapsto tx$, be a topological semiflow on a topological space $X$ with phase semigroup $T$. We introduce and discuss in this paper various transitivity dynamics of $(T,X)$.
In this paper, we study the topological structure of the Hahn field whose elements are functions from the additive abelian group of rational numbers to the real numbers field, with well-ordered support. After reviewing the algebraic and order structures of the Hahn field, we introduce different vector topologies that are induced by families of semi-norms and all of which are weaker than the order or valuation topology. We compare those vector topologies and we identify the weakest one whose properties are similar to those of the weak topology on the Levi-Civita field. In particular, we state a…
We show that for any discrete semigroup $X$ the semigroup operation can be extended to a right-topological semigroup operation on the space $G(X)$ of inclusion hyperspaces on $X$. We detect some important subsemigroups of $G(X)$, study the minimal ideal, the (topological) center, left cancelable elements of $G(X)$, and describe the structure of the semigroups $G(\IZ_n)$ for small numbers $n$.
We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_δ$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.
The interrelations between various classes of convergence spaces defined by countability conditions are studied. Remarkably, they all find characterizations in the usual space of ultrafilters in terms of classical topological properties. This is exploited to produce relevant examples in the realm of convergence spaces from known topological examples.
Topological embeddings into transformation monoids
P.J. Šafárik University in Košice（科希策帕·约·沙法里克大学）；TU Wien（维也纳工业大学）；Binghamton University（宾汉姆顿大学）；University of St Andrews（圣安德鲁斯大学）；University of Hertfordshire（赫特福德大学）
We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is computable if its boundary $\partial M$ is computable. We also show how this result combined with certain construction which compactifies a semicomputable set leads to the conclusion that some noncompact semicomputable manifolds in computable metric spaces are computable.
Given a topological ring $R$, we study semitopological $R$-modules, construct their completions, Bohr and borno modifications. For every topological space $X$, we construct the free (semi)topological $R$-module over $X$ and prove that for a $k$-space $X$ its free semitopological $R$-module is a topological $R$-module. Also we construct a Tychonoff space $X$ whose free semitopological $R$-module is not a topological $R$-module.
