Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

On locally countably generated differential spaces

On locally countably generated differential spaces DEMONSTRATE MATHEMATICAvol.XXINo41988Vieslaw SasinON LOCALLY COUNTABLY GENERATED DIFFERENTIAL SPACESDedicated to the memoryof Professor Edward OttoIn this paperwegiveacharacterizationofSikorski'sdifferential spaces [6] which are locally countably generated bya family of real-valued functions. In sectionIJpropertiesofthedifferentialspace1westudythe(R ,S^)ofallsequences with the natural differential structureS^generatedby projection [21. In Section 2 we describe somedifferential spaces whicharelocallypropertiesdiffeomorphicrealoftoopenLet R^ be the set of all real sequences. Every elementXCR"subsets of R N .1. R N as a differential spaceis a mapx: N — • Rfrom the set of natural numbersintotheset of real numbers. Ve will denote it by x =NDenote by n^ for i e N the projection of Rontothei-thcoordinate given by(1)n ^ x ) = x.for x = (x.) € R N .Let P = {n^: i c N} be the set of allTpu such projections. Bywe denote the weakest topology in R in which all projections- 897 -V. Sasinfrom P are continuous, i.e. the product topology.uLet S^ be the differential structure on Rgenerated bytheset P. By definition acS^ iff for any xefi''' there exists anneighbourhood V e Tp of x, natural numbers i^Vfunction (a : R• R such that(2)a(x) = a»(x.Thus aex. )i^openandaC00for x e V.iff locally it is a C® function of a finite number ofvariables.For any a e(3)and http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

On locally countably generated differential spaces

Sasin, Vieslaw
Demonstratio Mathematica , Volume 21 (4): 16 – Oct 1, 1988
Download PDF
16 pages

Loading next page...
 
/lp/de-gruyter/on-locally-countably-generated-differential-spaces-f3jQoP70Fr

References (3)

Publisher
de Gruyter
Copyright
© by Vieslaw Sasin
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1988-0407
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATE MATHEMATICAvol.XXINo41988Vieslaw SasinON LOCALLY COUNTABLY GENERATED DIFFERENTIAL SPACESDedicated to the memoryof Professor Edward OttoIn this paperwegiveacharacterizationofSikorski'sdifferential spaces [6] which are locally countably generated bya family of real-valued functions. In sectionIJpropertiesofthedifferentialspace1westudythe(R ,S^)ofallsequences with the natural differential structureS^generatedby projection [21. In Section 2 we describe somedifferential spaces whicharelocallypropertiesdiffeomorphicrealoftoopenLet R^ be the set of all real sequences. Every elementXCR"subsets of R N .1. R N as a differential spaceis a mapx: N — • Rfrom the set of natural numbersintotheset of real numbers. Ve will denote it by x =NDenote by n^ for i e N the projection of Rontothei-thcoordinate given by(1)n ^ x ) = x.for x = (x.) € R N .Let P = {n^: i c N} be the set of allTpu such projections. Bywe denote the weakest topology in R in which all projections- 897 -V. Sasinfrom P are continuous, i.e. the product topology.uLet S^ be the differential structure on Rgenerated bytheset P. By definition acS^ iff for any xefi''' there exists anneighbourhood V e Tp of x, natural numbers i^Vfunction (a : R• R such that(2)a(x) = a»(x.Thus aex. )i^openandaC00for x e V.iff locally it is a C® function of a finite number ofvariables.For any a e(3)and

Journal

Demonstratio Mathematica – de Gruyter

Published: Oct 1, 1988

There are no references for this article.