arXiv subjects
Publications and source records attributed to Julian Dontchev.
In 1990, Ganster and Reilly proved that a function is continuous if and only if it is precontinuous and LC-continuous. In this paper we extend their decomposition of continuity in terms of ideals. We show that a function $f \colon (X,τ,{\cal I}) \to (Y,σ)$ is continuous if and only if it is pre-I-continuous and I-LC-continuous. We also provide a decomposition of I-continuity.
The aim of this paper is introduce and initiate the study of extremally $T_1$-spaces, i.e., the spaces where all hereditarily compact $C_2$-subspaces are closed. A $C_2$-space is a space whose nowhere dense sets are finite.
A topological space $(X,τ)$ is called a locally LC-space if every point of $X$ has a neighborhood $U$ such that every Lindelöf subset of $(U,τ|U)$ is a closed subset of $(U,τ|U)$. The aim of this paper is to continue the study of locally LC-spaces.
We answer a recent question of Pyrih by proving that a topological space $(X,τ)$ is open-normal if and only if it is extremally disconnected.
The aim of this paper is to introduce the class of ${\cal A}{\cal B}$-sets as the sets that are the intersection of an open and a semi-regular set. Several classes of well-known topological spaces are characterized via the new concept. A new decomposition of continuity is provided.
This paper covers some recent progress in the study of sg-open sets, sg-compact spaces, N-scattered spaces and some related concepts. A subset $A$ of a topological space $(X,τ)$ is called sg-closed if the semi-closure of $A$ is included in every semi-open superset of $A$. Complements of sg-closed sets are called sg-open. A topological space $(X,τ)$ is called sg-compact if every cover of $X$ by sg-open sets has a finite subcover. N-scattered space is a topological spaces in which every nowhere dense subset is scattered.
The aim of this paper is to continue the study of sg-compact spaces. The class of sg-compact spaces is a proper subclass of the class of hereditarily compact spaces. In our paper we shall consider sg-compactness in product spaces. Our main result says that if a product space is sg-compact, then either all factor spaces are finite, or exactly one factor space is infinite and sg-compact and the remaining ones are finite and locally indiscrete.
Recently, Saleh claimed to have solved `a long standing open question' in Topology; namely, he proved that every almost continuous function is closure continuous (= $θ$-continuous). Unfortunately, this problem was settled long time ago and even a better result is known.
The aim of this survey article is to cover most of the recent research on preopen sets. I try to present majority of the results on preopen sets that I am aware of.
Recently, Mršević and Reilly discussed some covering properties of a topological space and its associated $α$-topology in both topological and bitopological ways. The main aim of this paper is to investigate some common and controversial covering properties of $\cal T$ and ${\cal T}^α$.
The aim of this paper is to introduce a new weak separation axiom that generalizes the separation properties between $T_1$ and completely Hausdorff. We call a topological space $(X,τ)$ a $T_{κ,ξ}$-space if every compact subset of $X$ with cardinality $\leq κ$ is $ξ$-closed, where $ξ$ is a general closure operator. We concentrate our attention mostly on two new concepts: kd-spaces and $T_{1/3}$-spaces.
An ideal is a nonempty collection of subsets closed under heredity and finite additivity. The aim of this paper is to unify some weak separation properties via topological ideals. We concentrate our attention on the separation axioms between $T_0$ and $T_2$. We prove that if $(X,τ,{\cal I})$ is a semi-Alexandroff $T_{\cal I}$-space and $\cal I$ is a $τ$-boundary, then $\cal I$ is completely codense.
The aim of this note is to show that every subset of a given topological space is the intersection of a preopen and a preclosed set, therefore $β$-locally closed, and that every topological space is $β$-submaximal.
The aim of this paper is to show that every scattered subset of a dense-in-itself semi-$T_D$-space is nowhere dense. We are thus able to answer a recent question of Coleman in the affirmative. In terms of Digital Topology, we prove that in semi-$T_D$-spaces with no open screen, trace spaces have no consolidations.
In this paper we will continue the study of p-closed spaces. This class of spaces is strictly placed between the class of strongly compact spaces and the class of quasi-H-closed spaces. We will provide new characterizations of p-closed spaces and investigate their relationships with some other classes of topological spaces.
The aim of this paper is to introduce and study the concept of a contra-semicontinuous function and further investigate the class of strongly $S$-closed spaces. We obtain some new decompositions of generalized continuous functions.
In this paper we define the concepts of $g.Λ_s$-sets and $g.V_s$-sets and we use them in order to obtain new characterizations of semi-T_1-, semi-R_0- and semi-T_{1/2}-spaces.
The aim of this paper is to continue the study of sg-compact spaces, a topological notion much stronger than hereditary compactness. We investigate the relations between sg-compact and $C_2$-spaces and the interrelations to hereditarily sg-closed sets.