Results on C2-paracompactness

A C-paracompact is a topological space X associated with a paracompact space Y and a bijective function f : X −→ Y satisfying that f A: A −→ f(A) is a homeomorphism for each compact subspace A ⊆ X. Furthermore, X is called C2-paracompact if Y is T2 paracompact. In this article, we discuss the above concepts and answer the problem of Arhangel’ski ̆i. Moreover, we prove that the sigma product Σ(0) can not be condensed onto a T2 paracompact space.

Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

We strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$ -compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to $X$ , and $\widehat{K}$ contains no analytic discs.

C-Tychonoff and L-Tychonoff Topological Spaces

A topological space X is called C-Tychonoff if there exist a one-to-one function f from X onto a Tychonoff space Y such that f restriction K from K onto f(K) is a homeomorphism for each compact subspace K of X. We discuss this property and illustrate the relationships between C-Tychonoffness and some other properties like submetrizability, local compactness, L-Tychononess, C-normality, C-regularity, epinormality, sigma-compactness, pseudocompactness and zero-dimensional.

C-normal topological property

A topological space X is called C-normal if there exist a normal space Y and a bijective function f : X ? Y such that the restriction f _ C : C ? f (C) is a homeomorphism for each compact subspace C ? X. We investigate this property and present some examples to illustrate the relationships between C-normality and other weaker kinds of normality.

EVERY COUNTABLE GROUP IS THE FUNDAMENTAL GROUP OF SOME COMPACT SUBSPACE OF

For every countable group $G$ we construct a compact path connected subspace $K$ of $\mathbb{R}^{4}$ such that ${\it\pi}_{1}(K)\cong G$. Our construction is much simpler than the one found recently by Virk.

Extremal problems for regenerative phenomena

This paper explores the possibility of a calculus of variations powerful enough to prove inequalities for the p-functions of regenerative phenomena such as that conjectured by Davidson and proved by Dai. It is shown that this is unlikely to be achieved by compactifying the space of standard p-functions, and a more promising approach is that of working in a compact subspace. The analysis leads to a class of candidate p-functions which contains all the maxima of general functionals.

Extremal problems for regenerative phenomena

This paper explores the possibility of a calculus of variations powerful enough to prove inequalities for the p-functions of regenerative phenomena such as that conjectured by Davidson and proved by Dai. It is shown that this is unlikely to be achieved by compactifying the space of standard p-functions, and a more promising approach is that of working in a compact subspace. The analysis leads to a class of candidate p-functions which contains all the maxima of general functionals.

A Note on Sarkovskiĭ's Theorem in Connected Linearly Ordered Spaces

We prove that, for a connected linearly ordered space L, the following conditions are equivalent: (1) L satisfies Sarkovskiĭ's Theorem, (2) there exist turbulent functions on L, and (3) there exists a compact subspace of L which satisfies Sarkovskiĭ's Theorem. Our results are applied in two ways. Firstly, we show that there exist connected linearly ordered spaces without infinite minimal sets; secondly, for each cardinal number λ of uncountable cofinality, we construct a connected linearly ordered space L such that: (1) L is a compact nonfirst countable space satisfying Sarkovskiĭ's Theorem, (2) L admits a dense first countable subset, and (3) the density of L is λ.