In this paper we continue with the study of paracompact maps introduced in [1]. We give two external characterizations for paracompact maps including a characterization analogous to The Tamano Theorem in the category TOP (of topological spaces and continuous maps as morphisms). A necessary and sufficient condition for the Tychonoff product of a closed map and a compact map to be closed is also ...

Preframes (directed complete posets with finite meets that distribute over the directed joins) are the algebras for an infinitary essentially algebraic theory, and can be presented by generators and relations. This result is combined with a general argument concerning categories of commutative monoids to give a very short proof of the localic Tychonoff Theorem. It is also shown how frames can b...

Extending the Stone Duality Theorem, we prove two duality theorems for category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. They extend also Tarski Theorem; latter is even derived from one them. We as well new EDTych extremally disconnected Tychonoff Also, describe categories which are dually equivalent to ZComp compactifications obtain a corollary Dwinger Theorem about space.

We give a new proof of the Komlós-Major-Tusnády embedding theorem for the simple random walk. The only external tool that we use is the Schauder-Tychonoff fixed point theorem for locally convex spaces. Besides that, the proof is almost entirely based on a series of soft arguments and easy inequalities, and no hard computations (implicit or explicit) are involved. This provides the first genuine...

In this paper we generalize the notion of perfect compactification of a Tychonoff space to a generic extension of any space by introducing the concept of perfect pair. This allow us to simplify the treatment in a basic way and in a more general setting. Some [S1], [S2], and [D]’s results are improved and new characterizations for perfect (Hausdorff) extensions of spaces are obtained.

Working in ZFC, we prove that for every zero-dimensional subspace S of the real line, the Tychonoff power ωS is homogeneous (ω denotes the nonnegative integers). It then follows as a corollary that ωS is homogeneous whenever S is a separable zero-dimensional metrizable space. The question of homogeneity in powers of this type was first raised by Ben Fitzpatrick, and was subsequently popularized...