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Some upper bounds for density of function spaces

Önal, Süleyman
Let C-alpha(X, Y) be the set of all continuous functions from X to Y endowed with the set-open topology where alpha is a hereditarily closed, compact network on X which is closed Under finite unions. We proved that the density of the space C-alpha(X, Y) is at most iw(X) . d(Y) where iw(X) denotes the i-weight of the Tychonoff space X, and d(Y) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function psi, and Y has a base consists of psi-convex Subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal kappa, there is a pathwise connected space Y Such that pi-weight of Y is kappa, but Souslin number of the Space C-kappa(vertical bar 0, 1 vertical bar, Y) is 2(kappa). (C) 2009 Elsevier B.V. All rights reserved.