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.

The category Top of topological spaces and continuous maps has the structures of a fibration category and a cofibration category in the sense of Baues, where fibration = Hurewicz fibration, cofibration = the usual cofibration, and weak equivalence = homotopy equivalence. Concentrating on fibrations, we consider the problem: given a full subcategory 𝓒 of Top, is the fibration structure of Top restricted to 𝓒 a fibration category? In this paper we take the special case where 𝓒 is the full subcategory...

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 $i$.

We first prove that given any analytic filter ℱ on ω the set of all functions f on $2^{\omega }$ which can be represented as the pointwise limit relative to ℱ of some sequence ${\left(fₙ\right)}_{n\in \omega }$ of continuous functions ($f=li{m}_{ℱ}fₙ$), is exactly the set of all Borel functions of class ξ for some countable ordinal ξ that we call the rank of ℱ. We discuss several structural properties of this rank. For example, we prove that any free Π⁰₄ filter is of rank 1.

We consider two situations which relate properties of filters with properties of the limit operators with respect to these filters. In the first one, we show that the space of sequences having limits with respect to a ${\Pi }_3^0$ filter is itself ${\Pi }_3^0$ and therefore, by a result of Dobrowolski and Marciszewski, such spaces are topologically indistinguishable. This answers a question of Dobrowolski and Marciszewski. In the second one, we characterize universally measurable filters which fulfill Fatou’s lemma.