Related papers: Discontinuous maps whose iterations are continuous

Let ${\mathscr P}$ be a topological property. We say that a space $X$ is ${\mathscr P}$-connected if there exists no pair $C$ and $D$ of disjoint cozero-sets of $X$ with non-${\mathscr P}$ closure such that the remainder $X\backslash(C\cup…

In this paper we search for conditions on a countably compact (pseudo-compact) topological semigroup under which: (i) each maximal subgroup $H(e)$ in $S$ is a (closed) topological subgroup in $S$; (ii) the Clifford part $H(S)$(i.e. the…

Knaster continua and solenoids are well-known examples of indecomposable continua whose composants (maximal arcwise-connected subsets) are one-to-one images of lines. We show that essentially all non-trivial one-to-one composant images of…

Let $C(\mathbf I)$ be the set of all continuous self-maps from ${\mathbf I}=[0,1]$ with the topology of uniformly convergence. A map $f\in C({\mathbf I})$ is called a transitive map if for every pair of non-empty open sets $U,V$ in…

A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\Phi:X\to exp(X)$ such that $[x,y]\subset\Phi(x)\cup \Phi(y)$ for all $x,y\in X$. We prove that each convex subset of…

Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point on X. We show that the set S consisting of all nonnegative integers n such that f^n(x) is in Y is a union of at most…

An action of a topological semigroup S on X is compactifiable if this action is a restriction of a jointly continuous action of S on a Hausdorff compact space Y. A topological semigroup S is compactifiable if the left action of S on itself…

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\mathbb{R}^2 \to \mathbb{R}^2$ preserving $f$, i.e. $f \circ h = f$. Denote by $S(f,r)$, $(0\leq r \leq \infty)$, the…

Let $X$ be a compact metric space and $T:X\longrightarrow X$ be continuous. Let $h^*(T)$ be the supremum of topological sequence entropies of $T$ over all subsequences of $\mathbb Z_+$ and $S(X)$ be the set of the values $h^*(T)$ for all…

The main goal of this note is to prove the following theorem. If $A_n$ is a sequence of measurable sets in a $\sigma$-finite measure space $(X, \mathcal{A}, \mu)$ that covers $\mu$-a.e. $x \in X$ infinitely many times, then there exists a…

Let $M$ be a compact Riemannian manifold. The set $\text{F}^{r}(M)$ consisting of sequences $(f_{i})_{i\in\mathbb{Z}}$ of $C^{r}$-diffeomorphisms on $M$ can be endowed with the compact topology or with the strong topology. A notion of…

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the…

A continous map $f: \mathbb{C}^n \rightarrow \mathbb{C}^N$ is $k$-regular if the image of any $k$ points spans a $k$-dimensional subspace. It is an important problem in topology and interpolation theory, going back to Borsuk and Chebyshev,…

A space $X$ is called {\it selectively pseudocompact} if for each sequence $(U_{n})_{n\in \mathbb{N}}$ of pairwise disjoint nonempty open subsets of $X$ there is a sequence $(x_{n})_{n\in \mathbb{N}}$ of points in $X$ such that $cl_X(\{x_n…

Given a set $X$ and a family $G$ of self-maps of $X$, we study the problem of the existence of a non-discrete Hausdorff topology on $X$ with respect to which all functions $f\in G$ are continuous. A topology on $X$ with this property is…

A topological group is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n=0,1,2,..., every locally aleph_n bounded topological…

Let $\mathcal{N} \neq \{0\}$ be a fixed set of integers, closed under multiplication, closed under negation, or containing $\{\pm 1\}$. We prove that any zero of a polynomial in $\mathbf{Z}[X]$ whose coefficients lie in $\mathcal{N}$ can be…

In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all…

We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…

For a separable locally compact but not compact metrizable space $X$, let $\alpha X = X \cup \{x_\infty\}$ be the one-point compactification with the point at infinity $x_\infty$. We denote by $EM(X)$ the space consisting of admissible…