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Property Q.

Bandy, C. (1991)

International Journal of Mathematics and Mathematical Sciences

Pseudo-homotopies of the pseudo-arc

Alejandro Illanes (2012)

Commentationes Mathematicae Universitatis Carolinae

Let X be a continuum. Two maps g , h : X → X are said to be pseudo-homotopic provided that there exist a continuum C , points s , t ∈ C and a continuous function H : X × C → X such that for each x ∈ X , H ( x , s ) = g ( x ) and H ( x , t ) = h ( x ) . In this paper we prove that if P is the pseudo-arc, g is one-to-one and h is pseudo-homotopic to g , then g = h . This theorem generalizes previous results by W. Lewis and M. Sobolewski.

Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces

Othman Echi (2003)

Bollettino dell'Unione Matematica Italiana

In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if g : Y → X is a quasi-homeomorphism, Z a sober space and f : Y → Z a continuous map, then there exists a unique continuous map F : X → Z such that F ∘ g = f . Let X be a T 0 -space, q : X → s X the injection of X onto its sobrification X s . It is shown, here, that q ⁢ Gold ⁢ X = Gold ⁢ X s , where Gold ⁢ X is the set of all locally closed points of X . Some applications are also indicated. The Jacobson prime spectrum...

Quasi-orbit spaces associated to T₀-spaces

C. Bonatti, H. Hattab, E. Salhi (2011)

Fundamenta Mathematicae

Let G ⊂ Homeo(E) be a group of homeomorphisms of a topological space E. The class of an orbit O of G is the union of all orbits having the same closure as O. Let E/G̃ be the space of classes of orbits, called the quasi-orbit space. We show that every second countable T₀-space Y is a quasi-orbit space E/G̃, where E is a second countable metric space. The regular part X₀ of a T₀-space X is the union of open subsets homeomorphic to ℝ or to 𝕊¹. We give a characterization of the spaces X with finite...

Raising dimension under all projections

John Cobb (1994)

Fundamenta Mathematicae

As a special case of the general question - “What information can be obtained about the dimension of a subset of ℝ n by looking at its orthogonal projections into hyperplanes?” - we construct a Cantor set in ℝ 3 each of whose projections into 2-planes is 1-dimensional. We also consider projections of Cantor sets in ℝ n whose images contain open sets, expanding on a result of Borsuk.

Recouvrements, derivation des mesures et dimensions.

Patrice Assouad, Thierry Quentin de Gromard (2006)

Revista Matemática Iberoamericana

Let X be a set with a symmetric kernel d (not necessarily a distance). The space (X,d) is said to have the weak (resp. strong) covering property of degree ≤ m [briefly prf(m) (resp. prF(m))], if, for each family B of closed balls of (X,d) with radii in a decreasing sequence (resp. with bounded radii), there is a subfamily, covering the center of each element of B, and of order ≤ m (resp. splitting into m disjoint families). Since Besicovitch, covering properties are known to be the main tool for...

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