Related papers: Left-Separating Order Types

We use a natural forcing to construct a left-separated topology on an arbitrary cardinal kappa. The resulting left-separated space X_kappa is also 0-dimensional T_2, hereditarily Lindelof, and countably tight. Moreover if kappa is regular…

Let $\kappa$ be an infinite regular cardinal. We define a topological space $X$ to be $T_{\kappa-Borel}$-space (resp. a $T_{\kappa-BP}$-space) if for every $x\in X$ the singleton $\{x\}$ belongs to the smallest $\kappa$-additive algebra of…

The present paper improves a result of V. Gutev and T. Nogura (1999) showing that a space $X$ is topologically well-orderable if and only if there exists a selection for $\mathcal{F}_2(X)$ which is continuous with respect to the Fell…

A topological space $X$ is a $\Delta$-space (or $X \in \Delta$) if for any decreasing sequence $\{A_n : n < \omega\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n : n < \omega\}$ of open sets with empty…

A coarse space $X$, endowed with a linear order compatible with the coarse structure of $X$, is called linearly ordered. We prove that every linearly ordered coarse space $X$ is locally convex and the asymptotic dimension of $X$ is either…

We classify the countable linear orders $X$ for which there is an order $A$ with at least two points such that the lexicographic product $AX$ is isomorphic to $X$. Given such an $X$, we determine every corresponding order $A$, and identify…

We show that a linearly ordered topological space is initially \lambda-compact if and only if it is \lambda-bounded, that is, every set of cardinality $\leq \lambda$ has compact closure. As a consequence, every product of initially…

A topological space $L$ is called a linear ordered topological space (LOTS) whenever there is a linear order $\leq$ on $L$ such that the topology on $L$ is generated by the open sets of the form $(a, b)$ with $a < b$ and $a, b \in L \cup \{…

We introduce and investigate a topological version of St\"ackel's 1907 characterization of finite sets, with the goal of obtaining an interesting notion that characterizes usual compactness (or a close variant of it). Define a $T_2$…

The pinning down number $ {pd}(X)$ of a topological space $X$ is the smallest cardinal $\kappa$ such that for any neighborhood assignment $U:X\to \tau_X$ there is a set $A\in [X]^\kappa$ with $A\cap U(x)\ne\emptyset$ for all $x\in X$.…

Given a cardinal $\kappa$ and a sequence $\left(\alpha_i\right)_{i\in\kappa}$ of ordinals, we determine the least ordinal $\beta$ (when one exists) such that the topological partition relation…

We prove that for every $T_0$ space $X$, there is a well-filtered space $W(X)$ and a continuous mapping $\eta_X: X\lra W(X)$ such that for any well-filtered space $Y$ and any continuous mapping $f: X\lra Y$ there is a unique continuous…

Every left-invariant ordering of a group is either discrete, meaning there is a least element greater than the identity, or dense. Corresponding to this dichotomy, the spaces of left, Conradian, and bi-orderings of a group are naturally…

A poset $\bfp$ is well-partially ordered (WPO) if all its linear extensions are well orders~; the supremum of ordered types of these linear extensions is the {\em length}, $\ell(\bfp)$ of $\bfp$. We prove that if the vertex set $X$ of…

We prove that, for every cardinal number $\alpha\geq {\mathfrak c}$, there exists a metrizable space $X$ with $|X|=\alpha$ such that for every pair of quasiorders $\leq_1$, $\leq_2$ on a set $Q$ with $|Q| \leq \alpha$ satisfying the…

We say that a Tychonoff space $X$ is a $\kappa$-space if it is homeomorphic to a closed subspace of $C_p(Y)$ for some locally compact space $Y$. The class of $\kappa$-spaces is strictly between the class of Dieudonn\'{e} complete spaces and…

All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $\kappa$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^\kappa$. Our main result shows…

We say that a (countably dimensional) topological vector space $X$ is orbital if there is $T\in L(X)$ and a vector $x\in X$ such that $X$ is the linear span of the orbit ${T^nx:n=0,1,...}$. We say that $X$ is strongly orbital if,…

Work of Linnell shows that the space of left-orderings of a group is either finite or uncountable, and in the case that the space is finite, the isomorphism type of the group is known---it is what is known as a Tararin group. By defining…

While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and…