### ${\aleph}_{1}$-directed inverse systems of continuous images of arcs.

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For every topological property $\mathcal{P}$, we define the class of $\mathcal{P}$-approximable spaces which consists of spaces X having a countable closed cover $\gamma $ such that the “section” $X(x,\gamma )=\bigcap \{F\in \gamma :x\in F\}$ has the property $\mathcal{P}$ for each $x\in X$. It is shown that every $\mathcal{P}$-approximable compact space has $\mathcal{P}$, if $\mathcal{P}$ is one of the following properties: countable tightness, ${\aleph}_{0}$-scatteredness with respect to character, $C$-closedness, sequentiality (the last holds under MA or ${2}^{{\aleph}_{0}}<{2}^{{\aleph}_{1}}$). Metrizable-approximable spaces are studied: every compact space in this class has...

It is known that for a nonempty topological space $X$ and a nonsingleton complete lattice $Y$ endowed with the Scott topology, the partially ordered set $[X,Y]$ of all continuous functions from $X$ into $Y$ is a continuous lattice if and only if both $Y$ and the open set lattice $\mathcal{O}X$ are continuous lattices. This result extends to certain classes of $\mathcal{Z}$-distributive lattices, where $\mathcal{Z}$ is a subset system replacing the system $\mathcal{D}$ of all directed subsets (for which the $\mathcal{D}$-distributive complete lattices are just the continuous...

A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.

Let X be a continuum. Let C(X) denote the hyperspace of all subcontinua of X. In this paper we prove that the following assertions are equivalent: (a) X is a dendroid, (b) each positive Whitney level in C(X) is 2-connected, and (c) each positive Whitney level in C(X) is ∞-connected (n-connected for each n ≥ 0).