We prove that every compact C1-submanifold of Rn, with or without boundary, has the bilipschitz extension property in Rn.

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of $tow\left(H_0\right)$ is an isomorphism if Y is movable. Recall that $\left(H_0\right)$ is the full subcategory of $pro-H_0$ consisting of...

Booleanization of frames or uniform frames, which is not functorial under the basic choice of morphisms, becomes functorial in the categories with weakly open homomorphisms or weakly open uniform homomorphisms. Then, the construction becomes a reflection. In the uniform case, moreover, it also has a left adjoint. In connection with this, certain dual equivalences concerning uniform spaces and uniform frames arise.

We show how Kirszbraun's theorem on extending Lipschitz mappings in Hilbert space implies its own generalization. There is a continuous extension operator preserving the Lipschitz constant of every mapping.

We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $\Sigma {⁰}_{\alpha }$, α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $\Pi {⁰}_{\alpha }$ uniformization which is the graph of a $\Sigma {⁰}_{\alpha }$-measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with $G_{\delta }$ sections.

We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections $E_x=y\in Y:(x,y)\in E$, x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.

Borsuk's quasi-equivalence relation on the class of all compacta is considered. The open problem concerning transitivity of this relation is solved in the negative. Namely, three continua X, Y and Z lying in ℝ³ are constructed such that X is quasi-equivalent to Y and Y is quasi-equivalent to Z, while X is not quasi-equivalent to Z.

The Borsuk-Sieklucki theorem says that for every uncountable family ${X_{\alpha }}_{\alpha \in A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $dim\left(X_{\alpha }\cap X_{\beta }\right)=n$. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $cl{c}_{\mathbb{Z}}^{n+1}$, where n ≥ 1, and G is an Abelian group. Let ${X_{\alpha }}_{\alpha \in J}$ be an uncountable family of closed subsets of X. If $di{m}_{G}X=di{m}_G{X}_{\alpha }=n$ for all α ∈ J, then $di{m}_G\left(X_{\alpha }\cap X_{\beta }\right)=n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...