Related papers: Orbital and strongly orbital spaces

A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\{U_x:x\in X\}$, there is a $D\subseteq X$ such that $\{U_x:x\in D\}$ covers $X$ and $D$ is locally finite in the topology generated by $\{U_x:x\in X\}$. We prove…

A bounded linear operator $T$ acting on a Banach space $\B$ is called weakly hypercyclic if there exists $x\in \B$ such that the orbit ${T^n x: n=0,1,...}$ is weakly dense in $\B$ and $T$ is called weakly supercyclic if there is $x\in \B$…

We give an affirmative answer to a question asked by Faghih-Ahmadi and Hedayatian regarding supercyclic vectors. We show that if $\mathcal X$ is an infinite-dimensional normed linear space and $T$ is a supercyclic operator on $\mathcal X$,…

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_\infty$-invariant and concentrated on a single…

A bounded linear operator T on Hilbert space is subspace-hypercyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace in a relatively dense set. We construct examples to show that subspace-hypercyclicity…

We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in $\R^n$ or non-positively curved n-dimensional…

Assume that $\mathcal{P}$ is a topological property of a space $X$, then we say that $X$ is {\it dense-$\mathcal{P}$} if each dense subset of $X$ has the property $\mathcal{P}$. In this paper, we mainly discuss dense subsets of a space $X$,…

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 topological space is nonseparably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space X is the image of a nonseparably connected complete…

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is $\ell_\infty$-barrelled, (ii) $L(X)$ is $\ell_\infty$-quasibarrelled, (iii) $L(X)$ is…

Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer…

A space X is called an alpha-Toronto space if X is scattered of Cantor-Bendixson rank alpha and is homeomorphic to each of its subspaces of same rank. We answer a question of Steprans by constructing a countable alpha-Toronto space for each…

Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y > x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic…

A topological space $X$ is called $\Cal A$-real compact, if every algebra homomorphism from $\Cal A$ to the reals is an evaluation at some point of $X$, where $\Cal A$ is an algebra of continuous functions. Our main interest lies on…

We characterize Ascoli spaces by showing that a Tychonoff space $X$ is Ascoli iff the canonical map from the free locally convex space $L(X)$ over $X$ into $C_k\big(C_k(X)\big)$ is an embedding of locally convex spaces. We prove that an…

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. If $X$ is Dieudonn\'{e} complete (for example, metrizable), then $L(X)$ is a reflexive group if and only if $X$ is discrete. Answering a question posed in [9] we prove…

According to Kim, Peris and Song, a continuous linear operator $T$ on a complex Banach space $X$ is called {\it numerically hypercyclic} if the numerical orbit $\{f(T^nx):n\in\N\}$ is dense in $\C$ for some $x\in X$ and $f\in X^*$…

A topological space $X$ is called almost discretely Lindel\"of if every discrete set $D \subset X$ is included in a Lindel\"of subspace of $X$. We say that the space $X$ is {\em $\mu$-sequential} if for every non-closed set $A \subset X$…

We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space X=(X,T) is countably compact iff it is countably subcompact relative…

In this paper, we prove that if $T$ is diskcyclic operator then the closed unit disk multiplied by the union of the numerical range of all iterations of $T$ is dense in $\mathcal H$. Also, if $T$ is diskcyclic operator and $|\lambda|\le 1$,…