Related papers: On maps preserving connectedness and /or compactne…

Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction…

Let $\F$ be a collection of subsets of $\Z_+$ and $(X,T)$ be a dynamical system. $x\in X$ is $\F$-recurrent if for each neighborhood $U$ of $x$, $\{n\in\Z_+:T^n x\in U\}\in \F$. $x$ is $\F$-product recurrent if $(x,y)$ is recurrent for any…

Let $X$ be metrizable, $Y$ be perfectly normal and suppose that there exists a uniformly continuous surjection $T: C_{p}(X) \to C_{p}(Y)$ (resp., $T: C_{p}^*(X) \to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the space of…

In this note, continuous transitive maps $f$ on the interval $I$ are re-addressed, where $I$ denotes one of the intervals: $(-\infty, \infty)$, $(-\infty, a]$, $[b, \infty)$, $[a, b]$, where $a < b$ are real numbers. Such maps must have a…

Let $S_{n}$ denote the space of all $n \times n$ real symmetric matrices. For n=2 or n>2 we characterize maps F from $S_{n}$ to $S_{m}$ which preserve adjacency, i.e. if rank(A-B)=1, then rank(F(A)-F(B))=1.

In a metric space, such as the real numbers with their standard metric, a set A is open if and only if no sequence with terms outside of A has a limit inside A. Moreover, a metric space is compact if and only if every sequence has a…

An additive map $T$ acting between spaces of vector-valued functions is said to be biseparating if $T$ is a bijection so that $f$ and $g$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. Note that an additive bijection retains…

In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function $f$ on a subset $E$ of $\R$, the set of real numbers is down continuous if it preserves downward half Cauchy sequences,…

We present a necessary and sufficient condition for a random product of maps on a compact metric space to be (strongly) synchronizing on average.

Let $H$ and $K$ be two complex inner product spaces with dim$(X)\geq 2$. We prove that for each non-zero additive mapping $A:H \to K$ with dense image the following statements are equivalent: $(a)$ $A$ is (complex) linear or…

The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…

We show that polynomial-time randomness (p-randomness) is preserved under a variety of familiar operations, including addition and multiplication by a nonzero polynomial-time computable real number. These results follow from a general…

Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and…

The attracting set and the inverse limit set are important objects associated to a self-map on a set. We call \emph{stable set} of the self-map the projection of the inverse limit set. It is included in the attracting set, but is not equal…

Brief development of the idea of the very important notion of continuity is given. Continuity is often confused with contiguity, "drawing the graph in one go," "no gaps," etc. The author argues in support of using correct notions of…

The 2-locality problem of diameter-preserving maps between C(X)-spaces is addressed in this paper. For any compact Hausdorff space X with at least three points, we give an example of a 2-local diameter-preserving map on C(X) which is not…

A topological space $Y$ has the property (B) of Banakh if there is a countable family $\{A_n:n\in \mathbb{N}\}$ of closed nowhere dense subsets of $Y$ absorbing all compact subsets of $Y$. In this note we show that the space $C_p(X)$ of…

Let X and Y be finite dimensional normed spaces, F(X,Y) a collection of all mappings from X into Y. A mapping $P\in F(X,Y)$ is said to be piecewise affine if there exists a finite family of convex polyhedral subsets covering X and such that…

Scaling-invariant functions preserve the order of points when the points are scaled by the same positive scalar (with respect to a unique reference point). Composites of strictly monotonic functions with positively homogeneous functions are…

Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other…