In this note we define a rather pathological connectedness property of Hausdorff spaces which is stronger than ordinary connectedness. We obtain a few basic properties of such spaces and derive a method for constructing them. It turns out that countable Hausdorff spaces having the connectedness property are as easily constructed as uncountable ones. Hence we have still another method for constr...

di(x,A) = inf{di(x,A) : a ∈ A}, i ∈ I Hi(A,B) = max{sup di(a,B), sup di(A, b)} = sup{|di(x,A) − di(x,B)|} where a ∈ A, b ∈ B and x ∈ X It is well known that on 2 , Hi is a pseudometric, called the induced Hausdorff pseudometric induced by di, i ∈ I. Let (X,U) be a unifrom space with an augumented associated family p∗. p∗ also induces a uniformity U∗ on 2 defined by the base β∗ = {V ∗(i, ε) : i ...

The Hausdorff distance, the Gromov-Hausdorff, the Fréchet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as infρ F (ρ) where F is a suitable functional and ρ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms...

135

In this paper we consider the problem that how large the Hausdorff dimension of E ⊂ Rd needs to be to ensure the radii set of (d − 1)-dimensional spheres determined by E has positive Lebesgue measure. We obtain two results. First, by extending a general mechanism for studying Falconer-type problems in [4], we prove that it holds when dimH(E) > d− 1+ 1 d and in R2, the index 3 2 is sharp for thi...

In this paper, the Urysohn and completely Hausdorff axioms in general topology are generalized to L-topological spaces so as to be compatible with pointwise metrics. Some properties and characterizations are also derived.

We establish a Quillen model structure on simplicial (symmetric) multicategories. It extends the model structure on simplicial categories due to J. Bergner [2]. We observe that our technique of proof enables us to prove a similar result for (symmetric) multicategories enriched over other monoidal model categories than simplicial sets. Examples include small categories, simplicial abelian groups...

The purpose of this paper is to investigate the phenomena that a sequence of Riemannian manifolds $M_{i}$ converges to ones with lower dimension, $N$, with respect to the Hausdorff distance, which is introduced in [11]. We have studied this phenomena in [7] and proved there that $M_{i}$ is a fibre bundle over $N$ with infranilmanifold fibre. In this paper, we study which fibre bundle it is, and...