The compact neighborhood extension property and local equi-connectedness

Uniform connectedness and uniform local connectedness for lattice-valued uniform convergence spaces

We apply Preuss' concept of $mbbe$-connectedness to the categories of lattice-valued uniform convergence spaces and of lattice-valued uniform spaces. A space is uniformly $mbbe$-connected if the only uniformly continuous mappings from the space to a space in the class $mbbe$ are the constant mappings. We develop the basic theory for $mbbe$-connected sets, including the product theorem. Furtherm...

Weak Banach-Saks property in the space of compact operators

For suitable Banach spaces $X$ and $Y$ with Schauder decompositions and‎ ‎a suitable closed subspace $mathcal{M}$ of some compact operator space from $X$ to $Y$‎, ‎it is shown that the strong Banach-Saks-ness of all evaluation‎ ‎operators on ${mathcal M}$ is a sufficient condition for the weak‎ ‎Banach-Saks property of ${mathcal M}$, where for each $xin X$ and $y^*in‎ ‎Y^*$‎, ‎the evaluation op...

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...

A connectedness property of algebraic moment maps

Let K be a connected compact Lie group and M a Hamiltonian K-manifold, i.e., a symplectic K-manifold equipped with a moment map μ : M → k := (LieK). A theorem of Kirwan (implicitly in [Ki]) asserts: if M is connected and compact then the level sets of μ are connected. The purpose of this note is to prove such a statement in the category of algebraic varieties. First, we reformulate Kirwan’s the...