A topological property is properly hereditary property if whenever every proper subspace has the property, the whole space has the property. In this note, we will study some topological properties that are preserved by proper subspaces; in fact, we will study the following topological properties: Baire spaces, second category, sequentially compact, hemicompact, δ-normal, and spaces having dispe...

Abstract Let $$K_1$$ K 1 , $$K_2$$ 2 be compact Hausdorff spaces and $$E_1, E_2$$ E , Banach not containing a copy of $$c_0$$ c 0 . We establish lower estimates t...

The α-modulation spaces M p,q , introduced by Gröbner in [10] are a class of function spaces that contain Besov spaces B p,q (α = 1) and modulation spaces M s p,q (α = 0) as special cases. There are two kinds of basic coverings on Euclidean R which is very useful in the theory of function spaces and their applications, one is the uniform covering R = ⋃ k∈Zn Qk, where Qk denote the unit cube wit...

A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1) points, with equality iff the unit ball is a parallelotope. We solve this conjecture in the affirmative for all 2-dimensional spaces and for spaces where the unit ball is...

A family of compact spaces containing continuous images of Radon-Nikodým compacta is introduced and studied. A family of Banach spaces containing subspaces of Asplund generated (i.e., GSG) spaces is introduced and studied. Further, for a continuous image of a Radon-Nikodým compact K we prove: If K is totally disconnected, then it is Radon-Nikodým compact. If K is adequate, then it is even Eberl...