On analyticity in cosmic spaces

Oleg Okunev

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 1, page 185-190
  • ISSN: 0010-2628

Abstract

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We prove that a cosmic space (= a Tychonoff space with a countable network) is analytic if it is an image of a K -analytic space under a measurable mapping. We also obtain characterizations of analyticity and σ -compactness in cosmic spaces in terms of metrizable continuous images. As an application, we show that if X is a separable metrizable space and Y is its dense subspace then the space of restricted continuous functions C p ( X ∣ Y ) is analytic iff it is a K σ δ -space iff X is σ -compact.

How to cite

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Okunev, Oleg. "On analyticity in cosmic spaces." Commentationes Mathematicae Universitatis Carolinae 34.1 (1993): 185-190. <http://eudml.org/doc/247471>.

@article{Okunev1993,
abstract = {We prove that a cosmic space (= a Tychonoff space with a countable network) is analytic if it is an image of a $K$-analytic space under a measurable mapping. We also obtain characterizations of analyticity and $\sigma $-compactness in cosmic spaces in terms of metrizable continuous images. As an application, we show that if $X$ is a separable metrizable space and $Y$ is its dense subspace then the space of restricted continuous functions $C_p(X\mid Y)$ is analytic iff it is a $K_\{\sigma \delta \}$-space iff $X$ is $\sigma $-compact.},
author = {Okunev, Oleg},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {measurable mapping; cosmic space; analyticity; topology of pointwise convergence; topology of pointwise convergence; cosmic space; measurable mapping; analyticity; separable metrizable space},
language = {eng},
number = {1},
pages = {185-190},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On analyticity in cosmic spaces},
url = {http://eudml.org/doc/247471},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Okunev, Oleg
TI - On analyticity in cosmic spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 1
SP - 185
EP - 190
AB - We prove that a cosmic space (= a Tychonoff space with a countable network) is analytic if it is an image of a $K$-analytic space under a measurable mapping. We also obtain characterizations of analyticity and $\sigma $-compactness in cosmic spaces in terms of metrizable continuous images. As an application, we show that if $X$ is a separable metrizable space and $Y$ is its dense subspace then the space of restricted continuous functions $C_p(X\mid Y)$ is analytic iff it is a $K_{\sigma \delta }$-space iff $X$ is $\sigma $-compact.
LA - eng
KW - measurable mapping; cosmic space; analyticity; topology of pointwise convergence; topology of pointwise convergence; cosmic space; measurable mapping; analyticity; separable metrizable space
UR - http://eudml.org/doc/247471
ER -

References

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  1. Arhangel'skiĭ A.V., Some topological spaces that arise in functional analysis, Russ. Math. Surveys 31 (1976), 14-30. (1976) MR0458366
  2. Arhangel'skiĭ A.V., Factorization theorems and function spaces: stability and monolithicity, Soviet. Math. Doklady 26 (1982), 177-181. (1982) 
  3. Arhangel'skiĭ A.V., Hurewicz spaces, analytic sets, and fan tightness of function spaces, Soviet. Math. Doklady 33 (1986), 396-399. (1986) 
  4. Christensen J.P.R., Topology and Borel Structure, North Holland, Amsterdam, 1974. Zbl0273.28001MR0348724
  5. Frolík Z., A measurable map with analytic domain and metrizable range is quotient, Bull. Amer. Math. Soc. 76 (1970), 1112-1117. (1970) MR0265539
  6. Kuratowski K., Topology, Vol. 1, Academic Press, N.Y.-London, 1966. Zbl0849.01044MR0217751
  7. Motorov D.B., Metrizable images of the arrow (Sorgenfrey line), Moscow Univ. Math. Bull. 39 (1984), 48-50. (1984) MR0741159
  8. Okunev O., On analyticity in non-metrizable spaces, Abstracts of the VII Prague Topol. Symp., p. 101. 
  9. Rogers C.A., Jayne J.E., and al., Analytic Sets, Academic Press, London, 1980. 
  10. Talagrand M., A new countably determined Banach space, Israel J. Math. 47 (1984), 75-80. (1984) Zbl0537.46019MR0736065

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