To Hopf’s conjecture about metric on topological product S2 × S2 of two 2-spheres

2018 ◽  
Author(s):  
Yu. A. Aminov
Keyword(s):  
Topological Product

scholarly journals On nearly S-paracompactness

2021 ◽  
Vol 20 ◽  
pp. 353-360
Author(s):  
José Sanabria ◽  
Osmin Ferrer ◽  
Clara Blanco
Keyword(s):  
Topological Product ◽  
Paracompact Spaces

The objective of the present work is to introduce the notion of α-nearly S-paracompact subset, which is closely related to α-nearly paracompact and αS-paracompact subsets. Moreover, we study the invariance under direct and inverse images of open, perfect and regular perfect functions of the nearly S-paracompact spaces [?] and analyze the behavior of such spaces through the sum and topological product

scholarly journals On exremality of affine image of topological product of some varieties

2021 ◽  
Vol 22 (2) ◽  
pp. 490-500
Author(s):  
Ilgar Shikar oglu Jabbarov ◽  
Leman Galib gizi Ismailova
Keyword(s):  
Topological Product ◽  
Affine Image

On the Compactification of Products

1971 ◽  
Vol 14 (4) ◽  
pp. 591-592 ◽  
Author(s):  
Christopher Todd
Keyword(s):  
Topological Product ◽  
Hausdorff Spaces ◽  
Completely Regular

Let {Xa, a ∊ A} be a family of completely regular Hausdorff spaces, {βXa} the corresponding family of their Stone-Čech compactifications and ΠaXa the usual topological product. The following theorem was proved by Glicksberg [2] and subsequently by Frolík [1].

scholarly journals Differentials of the 2nd kind on a product surface

1979 ◽  
Vol 2 (4) ◽  
pp. 615-626
Author(s):  
J. C. Wilson
Keyword(s):  
Topological Product ◽  
Product Surface ◽  
Complex Projective

This paper deals with the problems of representing an arbitrary double differential of the second kind, defined on a surface which is the topological product of two curves, in terms of the products of simple differentials of the second kind on the two curves. The curves are assumed to be non-singular and irreducible in a complex projective2-space.

The Structure of Continuous {0, 1}-Valued Functions on a Topological Product

1976 ◽  
Vol 28 (3) ◽  
pp. 553-559 ◽  
Author(s):  
S. Broverman
Keyword(s):  
Product Space ◽  
Discrete Space ◽  
Topological Product

In this paper we investigate the question of which continuous ﹛0, 1﹜-valued functions on a product space admit continuous extensions to where βXα is the Stone-Čech compactification of Xa and ﹛0, 1﹜ denotes the two point discrete space. This problem is clearly equivalent to determining which clopen subsets of have clopen closures in .

scholarly journals Free products of topological groups

1971 ◽  
Vol 4 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Free Product ◽  
Quotient Group ◽  
Closed Subgroup ◽  
Topological Product ◽  
Almost Periodic ◽  
Topological Groups ◽  
Free Products ◽  
Free Topological Group ◽  
Product Of Groups

In this note the notion of a free topological product Gα of a set {Gα} of topological groups is introduced. It is shown that it always exists, is unique and is algebraically isomorphic to the usual free product of the underlying groups. Further if each Gα is Hausdorff, then Gα is Hausdorff and each Gα is a closed subgroup. Also Gα is a free topological group (respectively, maximally almost periodic) if each Gα is. This notion is then combined with the theory of varieties of topological groups developed by the author. For a variety of topological groups, the -product of groups in is defined. It is shown that the -product, Gα of any set {Gα} of groups in exists, is unique and is algebraically isomorphic to the usual varietal product. It is noted that the -product of Hausdorff groups is not necessarily Hausdorff, but is if is abelian. Each Gα is a quotient group of Gα. It is proved that the -product of free topological groups of and projective topological groups of are of the same type. Finally it is shown that Gα is connected if and only if each Gα is connected.

Tychonoff's theorem in the framework of formal topologies

1997 ◽  
Vol 62 (4) ◽  
pp. 1315-1332 ◽  
Author(s):  
Sara Negri ◽  
Silvio Valentini
Keyword(s):  
Type Theory ◽  
Constructive Proof ◽  
Topological Product ◽  
Essential Difference ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
Definition Of ◽  
The Axiom Of Choice ◽  
Inductive Property ◽  
Formal Topology

In this paper we give a constructive proof of the pointfree version of Tychonoff's theorem within formal topology, using ideas from Coquand's proof in [7]. To deal with pointfree topology Coquand uses Johnstone's coverages. Because of the representation theorem in [3], from a mathematical viewpoint these structures are equivalent to formal topologies but there is an essential difference also. Namely, formal topologies have been developed within Martin Löf's constructive type theory (cf. [16]), which thus gives a direct way of formalizing them (cf. [4]).The most important aspect of our proof is that it is based on an inductive definition of the topological product of formal topologies. This fact allows us to transform Coquand's proof into a proof by structural induction on the last rule applied in a derivation of a cover. The inductive generation of a cover, together with a modification of the inductive property proposed by Coquand, makes it possible to formulate our proof of Tychonoff s theorem in constructive type theory. There is thus a clear difference to earlier localic proofs of Tychonoff's theorem known in the literature (cf. [9, 10, 12, 14, 27]). Indeed we not only avoid to use the axiom of choice, but reach constructiveness in a very strong sense. Namely, our proof of Tychonoff's theorem supplies an algorithm which, given a cover of the product space, computes a finite subcover, provided that there exists a similar algorithm for each component space.

The topological product structure of systems of Lebesgue spaces

1991 ◽  
Vol 290 (1) ◽  
pp. 527-543 ◽  
Author(s):  
Jan J. Dijkstra ◽  
Jerzy Mogilski
Keyword(s):  
Product Structure ◽  
Topological Product ◽  
Lebesgue Spaces
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