John Reynolds
We use convergence theory as the framework for studying H-closed spaces and H-sets in topological spaces. From this viewpoint, it becomes clear that the property of being H-closed and the property of being an H-set in a topological space are pretopological notions. Additionally, we define a version of H-closedness for pretopological spaces and investigate the properties of such a space.
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Tietze extension does not always work in constructive mathematics if closed sets are defined as sequentially closed sets
本文通过离散度量空间上的不可扩张算法函数构造反例，证明在构造性数学中当闭集定义为序列闭集时 Tietze 扩张定理不成立。
Here we have studied the ideas of g*-closed sets, g^tou -sets and Lamda*-closed sets and investigate some of their properties in the spaces of A. D. Alexandroff [1]. We have also studied few separation axioms like T-omega/4,T-3omega/8, T-omega in Alexandroff spaces and have introduced a new separation axiom namely T-5omega/8 axiom in this space.
A class of subsets designated as very thin subsets of natural numbers has been studied and seen that theory of convergence may be rediscovered if very thin sets are given to play main role instead of thin or finite sets which removes some drawback of statistical convergence. While developing the theory of very thin sets, concepts of super thin, very very thin and super super thin sets are evolved spontaneously.
Here we have investigated some aspects of $sλ$-closed sets on separation axioms including $s T_{2\frac{1}{2}} $ and $s T_{3\frac{1}{2}} $ axioms and on compactness in generalized topological spaces
The purpose of this paper is to continue studying the properties of $γ$-regular open sets introduced and explored in [6]. The concept of $γ$-closed spaces have also been defined and discussed.
On $a$-locally closed sets
本文借助 Ekici 的 $a$-开集提出 $a$-局部闭集概念，研究其性质与刻画，并阐明其与已有各类局部闭集的关系。
A remark on $β$-locally closed sets
本文的目的是证明给定拓扑空间的每个子集都是预开集和预闭集的交，因此是$β$ -局部闭的，并且每个拓扑空间都是$β$ -次极大的.
Here we have studied the ideas of $ sg_λ,sλ$ and $ sβ_λ$-closed sets and investigated some of their properties in generalized topological spaces. We have also studied some low separation axioms namely $ sλT_\frac{1}{4} $, $ sλT_\frac{3}{8} $, $ sλT_\frac{1}{2} $ axioms and their mutual relations with $ sλT_0 $ and $sλT_1 $ axioms.
Here we have introduced the ideas of $ (j-i)sg_κ^*$-closed sets and a semi generalized closed set in a bispace; $ i,j=1,2; i\not=j $ and then have studied on pairwise semi $T_0 $-axiom, pairwise semi $T_1 $-axiom and pairwise semi $T_ω$-axiom. We have investigated some of their topological properties and also established a relation among these axioms under some additional conditions.
Here we have introduced and studied the idea of $ Ig^*$-closed set with respect to an ideal and investigated some of its properties in Alexandroff spaces. We have also introduced $ Ig^*$-$T_0 $ axiom, $ Ig^*$-$T_1$ axiom, $ Ig^*$-$T_ω$ axiom and explored a relation among them
Closed sets in a generalization of the Alexandrov cube
本文继续研究由收敛 λ_ls 诱导的拓扑，重点利用极小元素族刻画闭集和闭包算子，并特别关注满足可数链条件的完备布尔代数。
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…
Here we have studied the idea of semi Lamda*-closed sets and investigate some of their properties in spaces considered by A. D. Alexandroff [1]. We have introduced some new separation axioms namely semi-Tw/4, , semi-T3w/8, semi-T5w/8, and their inter-relation with semi-T0 and semi-T1. We have shown that under certain conditions these axioms are equivalent.
The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all $closed$ neighborhoods of a point intersect C, this point is in C. We define a new topological cardinal invariant function, the $θ-bitighness small number$ of a space X, bts_theta(X), and prove that in every topological space X, the cardinality of the theta-closed hull of each set A is at most |A|^{bts_theta(X)}. Using this result, we synthesize all earlier results on bounds on the cardinality of theta-closed hulls. We provide applications to P-spaces and to the almost-Lindelof number…
C-open Sets on Topological Spaces
该研究提出拓扑空间中C-开集与C-闭集的概念，探究其基本性质，定义相关算子并厘清关系，还给出了基于这类集合的连续映射与紧空间并考察其性质。
Covering by discrete and closed discrete sets
假设基数$κ$是相对于空间$X$的\emph{small}，如果$κ<Δ(X)$，其中$Δ(X)$是$X$中非空开集的最小基数。证明了Baire度量空间不能被少量离散集覆盖，并给出了一些推广。我们展示了一个ZFC的例子，经常Baire $σ$ -空间和一个一致的例子，一个正常的Baire摩尔空间，可以覆盖的一个小数目的离散集。最后，我们对线性序空间做一些评论。
Connectivity degrees of complements of closed sets in continua
本文扩展了连续统中补集连通点与贫集的研究，引入闭集补集的连通度，并证明在 $n=1$ 且 $A$ 为贫集或单点集时新定义等价于已知非割点/集定义。
On closed P-sets with ccc in the space omega^*
证明了在余空间ω ^* 中不可能有ccc闭P-集。
Hyperspaces of Closed Limit Sets
研究了拓扑空间的闭极限子集集上的Michael拓扑和Fell拓扑。特别注意的是所有极大极限集的子族。
The concept of statistical convergence based on asymptotic density is introduced in this article through nets. Some possible extensions of classical results for statistical convergence of sequences are obtained in this article, with extensions to nets.
Unconditionally tau-Closed and tau-Algebraic Sets in Groups
定义了群中的无条件$τ$ -闭集和$τ$ -代数集族，它们是Markov定义的无条件闭集和无条件代数集的自然推广.找到了这两个族重合的一个充分条件，特别证明了这两个族在任意基数群中至多重合$τ$。这一结果推广了马尔可夫定理的重合无条件封闭和代数集在一个可数群（众所周知，他们可能是不同的不可数群）和Podewski定理的拓扑性的任何ungebunden群。
We prove that the countable product of lines contains a Borel linear subspace $L\ne\mathbb R^ω$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes of ``large'' sets and Kuczma--Ger classes in the topological vector spaces $\mathbb R^n$ for $n\le ω$.
On the convergence sets of operator sequences on spaces of homogeneous type
本文研究了在同构类型空间上算子序列的收敛集，通过特定条件下的刻画，应用于经典分析中的具体算子序列。
The main purpose of this paper is to study \emph{$e$-separable spaces}, originally introduced by Kurepa as $K_0'$ spaces; we call a space $X$ $e$-separable iff $X$ has a dense set which is the union of countably many closed discrete sets. We primarily focus on the behaviour of $e$-separable spaces under products and the cardinal invariants that are naturally related to $e$-separable spaces. Our main results show that the statement "there is a product of at most $\mathfrak c$ many $e$-separable spaces that fails to be $e$-separable'" is equiconsistent with the existence of a weakly compact card…
A Class of Groups in Which All Unconditionally Closed Sets are Algebraic
证明了在可数群的直积的某些子群中，作为无条件闭集的性质与作为代数集的性质是一致的。特别地，这些性质在所有的阿贝尔群中一致。
We prove that each non-separable completely metrizable convex subset of a Frechet space is homeomorphic to a Hilbert space. This resolves an old (more than 30 years) problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowoslki and Torunczyk, this result implies that each closed convex subset of a Frechet space is homemorphic to $[0,1]^n\times [0,1)^m\times l_2(k)$ for some cardinals $0\le n\leω$, $0\le m\le 1$ and $k\ge 0$.
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