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On spread and condensations

Authors:	A V Arhangelskii

Institution:	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)

Abstract:	A space has a property ${\mathcal {P}}$ strictly if every finite power of 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 is countable, then 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 is a topological group such that $s(C_{p}(G))\leq \omega$ , then is strictly hereditarily Lindelöf and strictly hereditarily separable.

Keywords:	Spread hereditary density condensation Lindel\"{o}f space function spaces topology of pointwise convergence small diagonal caliber

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