In [3], Kinoshita defined the notion of $f^{*}.p.p.$ and he proved that each compact AR has $f^{*}.p.p.$ In [4], Yonezawa gave some examples of not locally connected continua with f.p.p., but without $f^{*}.p.p.$ In general, for each n=1,2,..., there is an n-dimensional continuum $X_n$ with f.p.p., but without $f^{*}.p.p.$ such that $X_n$ is locally (n-2)-connected (see [4, Addendum]). In this note, we show that for each n-dimensional continuum X which is locally (n-1)-connected, X has f.p.p. if and only if X has $f^{*}.p.p.$

We study the behavior of the minimal tightness and functional tightness of topological spaces under the influence of the functor of the permutation degree. Analytically: a) We introduce the notion of $\tau$-open sets and investigate some basic properties of them. b) We prove that if the map $f:X\to Y$ is $\tau$-continuous, then the map $S{P}^{n}f:S{P}^{n}X\to S{P}^{n}Y$ is also $\tau$-continuous. c) We show that the functor $S{P}^n$ preserves the functional tightness and the minimal tightness of compacts. d) Finally, we give some facts and properties on $\tau$-bounded...

In this paper, the relationships between metric spaces and $g$-metrizable spaces are established in terms of certain quotient mappings, which is an answer to Alexandroff’s problems.

Solecki has shown that a broad natural class of $G_{\delta }$ ideals of compact sets can be represented through the ideal of nowhere dense subsets of a closed subset of the hyperspace of compact sets. In this note we show that the closed subset in this representation can be taken to be closed upwards.

We consider dynamical systems on a separable metric space containing at least two points. It is proved that weak topological mixing implies generic chaos, but the converse is false. As an application, some results of Piórek are simply reproved.

The main purpose of this paper is to prove some theorems concerning inverse systems and limits of continuous images of arcs. In particular, we shall prove that if X = {Xa, pab, A} is an inverse system of continuous images of arcs with monotone bonding mappings such that cf (card (A)) ≠ w1, then X = lim X is a continuous image of an arc if and only if each proper subsystem {Xa, pab, B} of X with cf(card (B)) = w1 has the limit which is a continuous image of an arc (Theorem 18).

Recall that a space $X$ is a c-semistratifiable (CSS) space, if the compact sets of $X$ are $G_{\delta }$-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a $T_2$-space $X$ is a k-c-semistratifiable space if and only if $X$ has a $g$ function which satisfies the following conditions: (1) For each $x\in X$, $\left\{x\right\}=\bigcap \left\{g\right(x,n):n\in \mathbb{N}\}$ and $g(x,n+1)\subseteq g(x,n)$ for each $n\in \mathbb{N}$. (2) If a...

Arhangel’skiǐ proved that if $X$ and $Y$ are completely regular spaces such that $C_p\left(X\right)$ and $C_p\left(Y\right)$ are linearly homeomorphic, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. In addition he proved the same result for compactness, $\sigma$-compactness and realcompactness. In this paper we prove that if $\phi :C_p\left(X\right)\to C_p\left(X\right)$ is a continuous linear surjection, then $Y$ is pseudocompact provided $X$ is and if $\phi$ is a continuous linear injection, then $X$ is pseudocompact provided $Y$ is. We also give examples that both statements do not hold...