Related papers: Lonely points revisited

A topological space $X$ is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product $X=\prod X_i$ of almost discrete spaces $X_i$ the space $C_p(X)$ of continuous real-valued…

For any left orderable group G, we recall from work of McCleary that isolated points in the space of left orderings correspond to basic elements in the free lattice ordered group over G. We then establish a new connection between the…

Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role…

Loneliness does not only have emotional implications on a person but also on his/her well-being. The study of loneliness has been challenging and largely inconclusive in findings because of the several factors that might correlate to the…

In the paper we study properties of the set of critical points for self-similar sets. We introduce simple condition that implies at most countably many critical values and we construct a self-similar set with uncountable set of critical…

We give a self-contained introduction to isolated points on curves and their counterpoint, parameterized points, that situates these concepts within the study of the arithmetic of curves. In particular, we show how natural geometric…

In this paper, we mainly discuss the class of charming spaces, which was introduced by A.V. Arhangel'skii in [Remainders of metrizable spaces and a generalization of Lindel\"of $\Sigma$-spaces, Fund. Math., 215(2011), 87-100]. First, we…

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.…

In this paper we consider the question of when the space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology is countable dense homogeneous. In particular, we focus on the case when $X$ is countable…

Given a set P of points on the plane, a polygon with vertices in P is said to be empty if it contains no element of P in its interior. We show that every set of n points in general position on the plane determines at least…

In this paper, we develop the notions of weak/directional monotonicity (developed by Sesma-Sara et al. in terms of the \emph{Kulisch-Miranker order}) and the notion of $ G $-monotonicity (introduced by Santiago et al. for $ [0,\!1]$) for…

Let f be a unimodal map of the interval with critical point c. If the orbit of c is not dense then most points in lim{[0,1],f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without…

Let (X,d) be a metric space of p-negative type. Recently I. Doust and A. Weston introduced a quantification of the p-negative type property, the so called gap {\Gamma} of X. This talk introduces some formulas for the gap {\Gamma} of a…

A topological space is called P_2 ( P_3, P_{<omega} ) if and only if it does not contain two (three, finitely many) uncountable open sets with empty intersection. We show that (i) there are 0-dimensional P_{<omega} spaces of size 2^omega,…

A fixed number of passengers independently board one of several buses uniformly at random. The lonely passenger problem is to prove that the probability of at least one passenger being the only one in their bus is increasing in the number…

In this paper, we define the spaces with a regular base at non-isolated points and discuss some metrization theorems. We firstly show that a space $X$ is a metrizable space, if and only if $X$ is a regular space with a $\sigma$-locally…

In this paper, we investigate the sobriety of weakly first-countable spaces and give some sufficient conditions that the Scott topologies of the open set lattices are sober. The main results are: (1) Let $P$ and $Q$ be two posets. If…

A novel perspective on defining black holes designed to be more broadly applicable outside of asymptotically flat spacetimes, in the context of classical general relativity, is presented, discussed, and characterized. The construction…

Let $\mathcal{I}$ be an analytic P-ideal [respectively, a summable ideal] on the positive integers and let $(x_n)$ be a sequence taking values in a metric space $X$. First, it is shown that the set of ideal limit points of $(x_n)$ is an…

This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface $M$ which are also isolated critical points of their restrictions to the boundary. This class of…