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Conditions on a topological space $X$ under which the space $C(X,\mathbb{R})$ of continuous real-valued maps with the Isbell topology $\kappa $ is a topological group (topological vector space) are investigated. It is proved that the…

Let $X$ be a Hausdorff compact space and $C(X)$ be the algebra of all continuous complex-valued functions on $X$, endowed with the supremum norm. We say that $C(X)$ is (approximately) $n$-th root closed if any function from $C(X)$ is…

We study groups of homeomorphic bijections on spaces that are finite unions of compact connected linearly ordered subsets. We prove that all such groups when endowed with the topology of point-wise convergence are topological groups. }

In this article, we continue our study of Baire one functions on a topological space $X$, denoted by $B_1(X)$ and extend the well known M. H. Stones's theorem from $C(X)$ to $B_1(X)$. Introducing the structure space of $B_1(X)$, it is…

We prove a new fixed - point result for the image Im(j) of any continuous function j from K to (K x K), where K is a compact convex subset of a Hausdorff locally convex space, provided that the projection of Im(j) to the first factor is…

The problem of deciding, given a complex variety X, a point x in X, and a subvariety Z of X, whether there is an automorphism of X mapping x into Z is proved undecidable. Along the way, we prove the undecidability of a version of Hilbert's…

In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that…

A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping $f : (X \times Y) \to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff…

Let $G$ be an abelian group, and $F$ a downward directed family of subsets of $G$. The finest topology $\mathcal{T}$ on $G$ under which $F$ converges to $0$ has been described by I.Protasov and E.Zelenyuk. In particular, their description…

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is…

For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi-Lipschitz homeomorphisms, with…

Let a group $G$ act properly discontinuously and cocompactly on a locally compact space $X$. A Hausdorff compact space $Z$ that contains $X$ as an open subspace has the perspectivity property if the action $G\curvearrowright X$ extends to…

We prove that if a continuous piecewise-smooth map on $\mathbb{R}^n$ is comprised of two linear functions, has a bounded orbit, and satisfies a certain non-degeneracy condition, then it has a fixed point. The result has important…

Let $X$ be a compact toric variety. Let $Hol$ denote the space of based holomorphic maps from $CP^1$ to $X$ which lie in a fixed homotopy class. Let $Map$ denote the corresponding space of continuous maps. We show that $Hol$ has the same…

We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological…

In this paper we study topological entropy and recurrence properties of non-autonomous dynamical system generated by a family of continuous self maps on a compact space X. Specially, we introduce the pseudo-entropy and…

Let $(W,S)$ be a Coxeter system whose graph is connected, with no infinite edges. A self-map $\tau$ of $W$ such that $\tau_{\sigma\theta}\in \{\tau_{\theta},\ \sigma\tau_{\theta}\}$ for all $\theta\in W$ and all reflections $\sigma$…

Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This…

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space $X$ and a continuous surjective map $T : X \to X$. By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does…

Let $X$ be a subset of a Hilbert space. We prove that if $v\colon X\to \mathbb{R}^m$ is such that \begin{equation*} \Big\lVert v(x)-\sum_{i=1}^m t_iv(x_i)\Big\rVert\leq \Big\lVert x-\sum_{i=1}^m t_ix_i\Big\rVert \end{equation*} for all…