The purpose of this work is to prove the following result: If a doubling metric measure space supports a weak (1, p)–Poincaré inequality with p sufficiently small, then annuli are almost quasiconvex. We also obtain estimates for the Hausdorff s–content and the diameter of the spheres. Mathematics Subject Classification (2000). Primary 46E35; Secondary 31C15.

This document is a summary of basic facts about the “standard” category of compactly generated spaces introduced by McCord [McC69] (often referred to in the literature as “compactly generated weak Hausdorff spaces”, or “weak Hausdorff k-spaces”). The main references I’ve used for this material are Gaunce Lewis’s thesis [Lew, App. A], and Neil Strickland’s note [Str09] (especially for material a...

We consider the uniform hyperspaces of a uniform space X, its associated Hausdorff space sX and its Hausdorff completion nX, and investigate the relationships between the functors s and n and the various hyperspace functors. Thus Theorem 1 says that there is a natural isomorphism between the functors TTE and TTETT, where ii'A'is the space of non-empty closed subsets of X, endowed with the Hausd...

In this paper we study evolution inclusions generated by time dependent convex subdifferentials, with the orientor field F depending on a parameter. Under reasonable hypotheses on the data, we show that the solution set S(λ) is both Vietoris and Hausdorff metric continuous in λ ∈ Λ. Using these results, we study the variational stability of a class of nonlinear parabolic optimal control problems.

For 0 < a < 1, let/a(x) = 2,1,2~ Ri(x) for 0 s$ x < 1, where {#,}£, is the sequence of Rademacher functions. We give a class of fa so that their graphs have Hausdorff dimension 2 —a. The result is closely related to the corresponding unsolved question for the Weierstrass functions.

in this paper we establish a characterization of abelian compact hausdorff semigroups with unique idempotent and ideal retraction property. we also introduce a function algebra on a semitopological semigroup whose associated semigroup compactification is universal withrespect to these properties.

In [5] Todorčević shows that there is no Hausdorff gap (A,B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A,B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A,B) if A is Σ2”, and equivalent to ∀r (א L[r] 1 < א1). We also consider real-valued games corresponding to Hausdorff gaps, and show that ADR for pointclass...

‎recently‎, ‎hussain et al.‎, ‎discussed the concept of $wt$-distance on a‎ ‎metric type space‎. ‎in this paper‎, ‎we prove some fixed‎ ‎point theorems for classes of contractive type multi-valued operators‎, ‎by using $wt$-distances in the setting of a complete metric type space‎. ‎these results generalize a result of feng and liu on multi-valued operators.

We introduce modal compact Hausdorff spaces as generalizations of modal spaces, and show these are coalgebras for the Vietoris functor on compact Hausdorff spaces. Modal compact regular frames and modal de Vries algebras are introduced as algebraic counterparts of modal compact Hausdorff spaces, and dualities are given for the categories involved. These extend the familiar Isbell and de Vries d...

We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-Löf random sequence. More generally, a sequence has effective dimension s if and only if it is coarsely similar to a weakly s-random sequence. Further, for any s ă t, every sequence of effective dimension s can be changed on density at most H ́1ptq ́H ́1psq of its bits to produce a sequen...