# Article

Full entry | PDF   (0.1 MB)
Keywords:
measurable mapping; cosmic space; analyticity; topology of pointwise convergence
Summary:
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.
References:
[1] Arhangel'skiĭ A.V.: Some topological spaces that arise in functional analysis. Russ. Math. Surveys 31 (1976), 14-30. MR 0458366
[2] Arhangel'skiĭ A.V.: Factorization theorems and function spaces: stability and monolithicity. Soviet. Math. Doklady 26 (1982), 177-181.
[3] Arhangel'skiĭ A.V.: Hurewicz spaces, analytic sets, and fan tightness of function spaces. Soviet. Math. Doklady 33 (1986), 396-399.
[4] Christensen J.P.R.: Topology and Borel Structure. North Holland, Amsterdam, 1974. MR 0348724 | Zbl 0273.28001
[5] Frolík Z.: A measurable map with analytic domain and metrizable range is quotient. Bull. Amer. Math. Soc. 76 (1970), 1112-1117. MR 0265539
[6] Kuratowski K.: Topology. Vol. 1, Academic Press, N.Y.-London, 1966. MR 0217751 | Zbl 0849.01044
[7] Motorov D.B.: Metrizable images of the arrow (Sorgenfrey line). Moscow Univ. Math. Bull. 39 (1984), 48-50. MR 0741159
[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. MR 0736065 | Zbl 0537.46019

Partner of