Let L be finite CW complex. By [L] we denote extension type of L. The following generalization of Michael’s selection theorem is proved: Theorem. Consider [L] ≤ [S]. Let F be a lower semi-continuous map between polish space X and metrizable compactum Y , such that ed(X) ≤ [L], F is equi-LC collection and F (x) ∈ AE([L]) for any x ∈ X. Let A be a closed subset of X such that there exists a conti...

We show that if the space `∞/c0 contains an isometric copy of every function space over a first countable compactum or every function space over a Corson copactum of weight not exceeding the continuum then every subset of R belongs to the σfield generated by sets of the form A1×A2.We prove a similar result about isomorphic rather than isometric embeddings into `∞/c0 in terms of the σ-field of s...

Consider a system ẋ = f(x) + g(x)u (f and g locally Lipschitz) with the property that the associated autonomous system ż = f(z) has an asymptotically stable compactum C with region of attraction A. Assume that x is a solution of the former, defined on [0,∞), corresponding to an input function u. A simple proof is given of the following L -input converging-state property: if u ∈ L for some p ∈ [...

A compactum X is an ‘absolute cone’ if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for n ≤ 3 and false for n ≥ 5. For n = 4, the absolute...

We rephrase Gromov’s definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum X, dimZ(p) X = dimQ X for all but finitely many primes p where Z(p) is the localization of Z at p. We construct Markov compacta of arbitrarily large dimension having dimQ X = 1 as well ...

A homeomorphism f : X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x 6= y, then there is an integer n ∈ Z such that d(f(x), f(y)) > c. In this paper, we prove that if a homeomorphism f : X → X of a continuum X can be lifted to an onto map h : P → P of the pseudoarc P , then f is not expansive. As a corollary, we prove that there are no expansive ho...

Abstract The paper alluded to in the title contains following striking result: Let $I$ be unit interval and $\Delta$ Cantor set. If $X$ is a quasi Banach space containing no copy of $c_{0}$ which isomorphic closed subspace with basis $C(I,\,X)$ linearly homeomorphic $C(\Delta ,\, X)$ , then locally convex, i.e., space. We will show that Kalton result sharp by exhibiting non-locally convex space...

Origanum compactum Benth. has been widely used in moroccan traditional medicine for various therapeutic treatments. Belonging to the same genus, O. onites was found have marked analgesic and anti-inflammatory activities. The aim of this work is evaluate theses pharmacological properties essential oil order provide a basis folkloric use plant. Aerial parts plant were subjected steam distillation...