On a question of Borsuk concerning non-continuous retracts II

On a Question concerning Sharp Bases

A sharp base B is a base such that whenever (Bi)i<ω is an injective sequence from B with x ∈ i<ω Bi, then { ⋂ i<n Bi : n < ω} is a base at x. Alleche, Arhangel’skĭı and Calbrix asked: if X has a sharp base, must X × [0, 1] have a sharp base? Good, Knight and Mohamad claimed to construct an example of a Tychonoff space P with a sharp base such that P × [0, 1] does not have a sharp base. However,...

On a Question of Wilf concerning Numerical Semigroups

Let S be a numerical semigroup with embedding dimension e(S), Frobenius number g(S), and type t(S). Put n(S) := Card(S ∩ {0, 1, . . . , g(S)}). A question of Wilf is shown to be equivalent to the statement that e(S)n(S) ≥ g(S)+1. This question is answered affirmatively if S is symmetric, pseudo-symmetric, or of maximal embedding dimension. The question is also answered affirmatively in the foll...

On a question of Gowers concerning isosceles right-angle triangles

On a question of Gowers concerning isosceles right-angle triangles

Extension of the Borsuk Theorem on Non-embeddability of Spheres

It is proved that the suspension P M of a closed n-dimensional manifold M , n ≥ 1, does not embed in a product of n + 1 curves. In fact, the ultimate result will be proved in a much more general setting. This is a far-reaching generalization the Borsuk theorem on non-embeddability of the sphere Sn+1 in a product of n + 1 curves.