In previous papers, various notions of pre-Hausdorff, Hausdorff and regular objects at a point $p$ in a topological category were introduced and compared. The main objective of this paper is to characterize each of these notions of pre-Hausdorff, Hausdorff and regular objects locally in the category of proximity spaces. Furthermore, the relationships that arise among the various $\mathrm{Pre}{T}_2$, $T_i$, $i=0,1,2,3$, structures at a point $p$ are investigated. Finally, we examine the relationships between the generalized separation...

A topological space $X$ is totally Brown if for each $n\in \mathbb{N}\setminus \left\{1\right\}$ and every nonempty open subsets $U_1,U_2,...,U_n$ of $X$ we have ${\mathrm{cl}}_X\left(U_1\right)\cap {\mathrm{cl}}_X\left(U_2\right)\cap \cdots \cap {\mathrm{cl}}_X\left(U_n\right)\ne \varnothing$. Totally Brown spaces are connected. In this paper we consider a topology ${\tau }_S$ on the set $\mathbb{N}$ of natural numbers. We then present properties of the topological space $(\mathbb{N},{\tau }_S)$, some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014, 2016 and by...

In this paper the following two propositions are proved: (a) If $X_{\alpha }$, $\alpha \in A$, is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product ${\square }_{\alpha \in A}{X}_{\alpha }$ can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If $X_{\alpha }$, $\alpha \in A$, is an infinite system of Brown Hausdorff topological spaces then the box product ${\square }_{\alpha \in A}{X}_{\alpha }$ is also Brown Hausdorff, and hence, it is connected. A space is Brown if...

In [1], various generalizations of the separation properties, the notion of closed and strongly closed points and subobjects of an object in an arbitrary topological category are given. In this paper, the relationship between various generalized separation properties as well as relationship between our separation properties and the known ones ([4], [5], [7], [9], [10], [14], [16]) are determined. Furthermore, the relationships between the notion of closedness and strongly closedness are investigated...

A topological space $X$ is totally Brown if for each $n\in \mathbb{N}\setminus \left\{1\right\}$ and every nonempty open subsets $U_1,U_2,...,U_n$ of $X$ we have ${\mathrm{cl}}_X\left(U_1\right)\cap {\mathrm{cl}}_X\left(U_2\right)\cap \cdots \cap {\mathrm{cl}}_X\left(U_n\right)\ne \varnothing$. Totally Brown spaces are connected. In this paper we consider the Golomb topology ${\tau }_G$ on the set $\mathbb{N}$ of natural numbers, as well as the Kirch topology ${\tau }_K$ on $\mathbb{N}$. Then we examine subsets of these spaces which are totally Brown. Among other results, we characterize the arithmetic progressions which are either totally Brown or totally separated in $(\mathbb{N},{\tau }_G)$. We also show that $(\mathbb{N},{\tau }_G)$ and $(\mathbb{N},{\tau }_K)$ are aposyndetic. Our results...