This is to recollect the equivalent formulations of Brouwer fixed point theorem. We collect a large number recently known sources such equivalents. More recently, Jinlu Li obtained two theorems on newly defined quasi-point-separable topological vector spaces. His extend Tychonoff theorem locally convex t.v.s. However, we note that his new are logically Consequently, add up our list

Abstract We study the question for which Tychonoff spaces X and locally convex E space of continuous ‐valued functions on contains a complemented copy , both endowed with pointwise topology. provide positive answer vast class spaces, extending classical theorems Cembranos, Freniche, Domański Drewnowski, proved case Banach Fréchet . Also, given infinite Y we show that if only any such subspace.

In a recent paper by D. Shakhmatov and J. Spěvák [Group-valued continuous functions with the topology of pointwise convergence, Topology and its Applications (2009), doi:10.1016/j.topol.2009.06.022] the concept of a TAP group is introduced and it is shown in particular that the NSS groups are TAP. We prove that conversely, the Weil complete metrizable TAP groups are NSS. We define also the narr...

The notion of ultrametrics can be considered as a zero-dimensional analogue ordinary metrics, and it is expected to prove ultrametric versions theorems on metric spaces. In this paper, we provide the Arens-Eells isometric embedding theorem spaces, Hausdorff extension Niemytzki-Tychonoff characterization compactness, author’s interpolation metrics dense subsets spaces metrics.

‎let ${x_{alpha}:alphainlambda}$ be a collection of topological spaces‎, ‎and $mathcal {g}_{alpha}$ be a grill on $x_{alpha}$ for each $alphainlambda$‎. ‎we consider tychonoffrq{}s type theorem for $x=prod_{alphainlambda}x_{alpha}$ via the above grills and a natural grill on $x$ ‎related to these grills, and present a simple proof to this theorem‎. ‎this immediately yields the classical theorem...

The purpose of this work is to introduce and study a new topological property called epi-complete-regularity. A space (X, T ) an epi-completely-regular if there exists topology T′ on X which coarser than such that T′) Tychonoff. This investigated some examples are presented in illustrate its relationships with other kinds normality complete-regularity.

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X XX is not pseudocompact; (2) [ZFC] The product of any set of pse...

This paper investigates a new mathematical model of static frictional contact for an ideally locking piezoelectric material with electrical and thermal conductivity conditions. A variational formulation the is derived, in form coupled system displacements, electric potential temperature. The existence unique weak solution established its proof based on Tychonoff fixed point theorem multivalued ...

Every first countable pseudocompact Tychonoff space X has the property that every pseudocompact subspace of X is a closed subset of X (denoted herein by “FCC”). We study the property FCC and several closely related ones, and focus on the behavior of extension and other spaces which have one or more of these properties. Characterization, embedding and product theorems are obtained, and some exam...