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On spread and condensations

Author: A. V. Arhangelskii
Journal: Proc. Amer. Math. Soc. 124 (1996), 3519-3527
MSC (1991): Primary 54A25, 54C35, 54A35
MathSciNet review: 1353369
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Abstract: A space $X$ has a property ${\mathcal {P}}$ strictly if every finite power of $X$ has ${\mathcal {P}}$. A condensation is a one-to-one continuous mapping onto. For Tychonoff spaces, the following results are established. If the strict spread of $X$ is countable, then $X$ can be condensed onto a strictly hereditarily separable space. If $s(C_{p}(X))\leq \omega $, then $C_{p}(X)$ can be condensed onto a strictly hereditarily separable space, and therefore, every compact subspace of $C_{p}(X)$ is strictly hereditarily separable. Under $(MA+\neg CH)$, if $G$ is a topological group such that $s(C_{p}(G))\leq \omega $, then $G$ is strictly hereditarily Lindelöf and strictly hereditarily separable.

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Additional Information

A. V. Arhangelskii
Affiliation: Chair of General Topology and Geometry, Mech.-Math. Faculty, Moscow University, Moscow 119899, Russia (June 15–December 31); Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701 (January 1–June 15)

Keywords: Spread, hereditary density, condensation, Lindel\"{o}f space, function spaces, topology of pointwise convergence, small diagonal, caliber
Received by editor(s): April 7, 1995
Additional Notes: The author was partially supported by NSF grant DMS-9312363.
Communicated by: Franklin D. Tall
Article copyright: © Copyright 1996 American Mathematical Society