Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes $K$ is the implication $X\tau_h K\implies \beta(X)\tau K$ for all paracompact spaces $X$. Here are the main results of the paper: Theorem: If $\{K_s\}_{s\in S}$ is a family of pointed quasi-finite complexes, then their wedge $\bigvee\limits_{s\in S}K_s$ is quasi-finite. Theorem: If $K_1$ and $K_2$ are quasi-finite countable complexes, then their join $K_1\ast K_2$ is quasi-finite. Theorem: For every quasi-finite CW complex $K$ there is a family $\{K_s\}_{s\in S}$ of countable CW complexes such that $\bigvee\limits_{s\in S} K_s$ is quasi-finite and is equivalent, over the class of paracompact spaces, to $K$. Theorem: Two quasi-finite CW complexes $K$ and $L$ are equivalent over the class of paracompact spaces if and only if they are equivalent over the class of compact metric spaces. Quasi-finite CW complexes lead naturally to the concept of $X\tau {\mathcal F}$, where ${\mathcal F}$ is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept.Comment: 20 page

An approach to basic set theory and logic

The purpose of this paper is to outline a simple set of axioms for basic set theory from which most fundamental facts can be derived. The key to the whole project is a new axiom of set theory which I dubbed "The Law of Extremes". It allows for quick proofs of basic set-theoretic identities and logical tautologies, so it is also a good tool to aid one's memory. I do not assume any exposure to euclidean geometry via axioms. Only an experience with transforming algebraic identities is required. The idea is to get students to do proofs right from the get-go. In particular, I avoid entangling students in nuances of logic early on. Basic facts of logic are derived from set theory, not the other way around.Comment: 22 page

Isomorphisms in pro-categories

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. In \cite{DR2} we gave characterizations of monomorphisms (resp. epimorphisms) in arbitrary pro-categories, pro-(C), where (C) has direct sums (resp. weak push-outs). In this paper we introduce the notions of strong monomorphism and strong epimorphism. Part of their significance is that they are preserved by functors. These notions and their characterizations lead us to important classical properties and problems in shape and pro-homotopy. For instance, strong epimorphisms allow us to give a categorical point of view of uniform movability and to introduce a new kind of movability, the sequential movability. Strong monomorphisms are connected to a problem of K.Borsuk regarding a descending chain of retracts of ANRs. If (f: X \to Y) is a bimorphism in the pointed shape category of topological spaces, we prove that (f) is a weak isomorphism and (f) is an isomorphism provided (Y) is sequentially movable and $X$ or $Y$ is the suspension of a topological space. If (f: X \to Y) is a bimorphism in the pro-category pro-(H_0) (consisting of inverse systems in (H_0), the homotopy category of pointed connected CW complexes) we show that (f) is an isomorphism provided (Y) is sequentially movable.Comment: to appear in the Journal of Pure and Applied Algebr