For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^X$ can be extended to a Whitney map for $2^X$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.

We present a categorical approach to the extension of probabilities, i.e. normed $\sigma$-additive measures. J. Novák showed that each bounded $\sigma$-additive measure on a ring of sets $𝔸$ is sequentially continuous and pointed out the topological aspects of the extension of such measures on $𝔸$ over the generated $\sigma$-ring $\sigma \left(𝔸\right)$: it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space $X$ over its Čech-Stone compactification $\beta X$ (or as the extension of continuous...

Let $𝒟$ denote a true dimension function, i.e., a dimension function such that $𝒟\left({ℝ}^n\right)=n$ for all $n$. For a space $X$, we denote the hyperspace consisting of all compact connected, non-empty subsets by $C\left(X\right)$. If $X$ is a countable infinite product of non-degenerate Peano continua, then the sequence ${\left({𝒟}_{\ge n}\left(C\left(X\right)\right)\right)}_{n=2}^{\infty }$ is $F_{\sigma }$-absorbing in $C\left(X\right)$. As a consequence, there is a homeomorphism $h:C\left(X\right)\to Q^{\infty }$ such that for all $n$, $h\left[\{A\in C\left(X\right):𝒟\left(A\right)\ge n+1\}\right]=B^n\times Q\times Q\times \cdots$, where $B$ denotes the pseudo boundary of the Hilbert cube $Q$. It follows that if $X$ is a countable infinite product of non-degenerate...

An embedding from a Cartesian product of two spaces into the Cartesian product of two spaces is said to be factorwise rigid provided that it is the product of embeddings on the individual factors composed with a permutation of the coordinates. We prove that each embedding of a product of two pseudo-arcs into itself is factorwise rigid. As a consequence, if X and Y are metric continua with the property that each of their nondegenerate proper subcontinua is homeomorphic to the pseudo-arc, then X ×...

A continuum is a compact connected metric space. For a continuum X, let C(X) denote the hyperspace of subcontinua of X. In this paper we construct two nonhomeomorphic fans (dendroids with only one ramification point) X and Y such that C(X) and C(Y) are homeomorphic. This answers a question by Sam B. Nadler, Jr.

We study an order relation on the fibers of a continuous map and its application to the study of the structure of compact spaces of uncountable weight.

We effectively construct in the Hilbert cube $ℍ={[0,1]}^{\omega }$ two sets $V,W\subset ℍ$ with the following properties: (a) $V\cap W=\varnothing$, (b) $V\cup W$ is discrete-dense, i.e. dense in ${{[0,1]}_D}^{\omega }$, where ${[0,1]}_D$ denotes the unit interval equipped with the discrete topology, (c) $V$, $W$ are open in $ℍ$. In fact, $V={\bigcup }_{\mathbb{N}}{V}_i$, $W={\bigcup }_{\mathbb{N}}{W}_i$, where $V_i={\bigcup }_0^{2^{i-1}-1}{V}_{ij}$, $W_i={\bigcup }_0^{2^{i-1}-1}{W}_{ij}$. $V_{ij}$, $W_{ij}$ are basic open sets and $(0,0,0,...)\in V_{ij}$, $(1,1,1,...)\in W_{ij}$, (d) $V_i\cup W_i$, $i\in \mathbb{N}$ is point symmetric about $(1/2,1/2,1/2,...)$. Instead of $[0,1]$ we could have taken any $T_4$-space or a digital interval, where the resolution (number of points) increases with $i$.