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Let $X$ be a continuum. In Proposition 31 of J.J. Charatonik and W.J. Charatonik, , Comment. Math. Univ. Carolin. (2000), no. 1, 123–132, it is claimed that $L\left(X\right)={\bigcap }_{p\in X}S\left(p\right)$, where $L\left(X\right)$ is the set of points at which $X$ is locally connected and, for $p\in X$, $a\in S\left(p\right)$ if and only if $X$ is smooth at $p$ with respect to $a$. In this paper we show that such equality is incorrect and that the correct equality is $P\left(X\right)={\bigcap }_{p\in X}S\left(p\right)$, where $P\left(X\right)$ is the set of points at which $X$ is connected im kleinen. We also use the correct equality to obtain some results concerning...

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...

We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions $P(a,b)$ with the property that every prime number that divides $a$ also divides $b$, it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are aposyndetic...

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...