We investigate whether an arbitrary base for a topological space can be partitioned into two bases. We prove that every base for a T3 Lindelöf topology can be partitioned into two bases while there exists a consistent example of a rst countable, 0-dimensional, Hausdor space of size and weight ω1 which admits a base without a partition to two bases.

Ametric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces [1], we show that a countable ultrametric space embeds isometrically into an indivisible ultrametric metric space if and only if it does not contain a strictly increasing sequence of balls.

Assuming Martin’s Axiom and א1 < 2 א0 , we show that, for any κ, λ < 20 and any almost disjoint family {ai : i < λ} of countable subsets of κ, there is a partition {pn : n ∈ ω} of κ so that pn ∩ ai is finite for each 〈i, n〉 ∈ λ × ω.

Let T be a ,nite set of ,nite tournaments and HT the countable homogeneous directed graph which does not embed any of the tournaments in T. We will completely describe the vertex partition property HT. c © 2002 Elsevier Science B.V. All rights reserved.

For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space Zα such that indZα = α, and no closed subset L of Zα with indL less than the predecessor of α is a partition in Zα. An α-dimensional Cantor Ind-manifold can be constructed similarly.

In this note we shall characterize a topological property which is stronger than countable paracompactness but which is equivalent to it for normal spaces. A real valued function on a topological space X is locally bounded if each point has a neighborhood on which the function is bounded. Let C(X) denote the set of real valued continuous functions on X. A topological space is a cb-space if for ...

This paper focuses on completeness results about generic expansions of propositional Weak Nilpotent Minimum (WNM) logics with truth-constants. Indeed, we consider algebraic semantics for expansions of these logics with a set of truth-constants {r | r ∈ C}, for a suitable countable C ⊆ [0, 1], and provide a full description of completeness results when (i) the t-norm is a Weak Nilpotent Minimum ...

The assumption of [HM] and our assumption here, the existence of a strong partition cardinal, is moderately special. On the one hand, it violates the Axiom of Choice and is not relatively consistent with ZF (unlike AC and its negation). On the other hand, under the Axiom of Determinacy (AD), such cardinals are abundant and consistent with countable choice and DC, the principle of Dependent Choi...

A set HE w is said to be diverse with respect to a partition II of w if at least two pieces of I7 have an infinite intersection with H. A family of partitions of w has the Ramsey property if, whenever Co] is two-colored, some monochromatic set is diverse with respect to at least one partition in the family. We show that no countable collection of even infinite partitions of CIJ has the Ramsey p...

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The gives framework number of results in the literature concerning models which independence i is strictly below c, including iterations Sacks forcing, Miller partition h-perfect tree forcings, coding with perfect trees. Moreover, applying theorem, we show that i=ℵ1 Lite m...