Topological classification of function spaces with the Fell topology IV

摘要

For a Tychonoff space X, let down arrow C-F (X) denote the collection of the hypographs of all continuous maps from X to [0, 1] with the Fell topology. We show that, for a Tychonoff non-discrete k-space X, the function space down arrow C-F (X) is homeomorphic to c(0) boolean OR (Q Sigma) if down arrow C-F (X) is metrizable and the set of isolated points of X is dense in X, where Q = [-1, 1](N) is the Hilbert cube, Sigma = {(x(n)) is an element of Q : sup |x(n)| < 1} and c(0) = {(x(n)) is an element of Sigma : lim x(n) = 0} are its subspaces. Combining results in the previous papers of the series, we give the topological classification for all metrizable function spaces (X) of down arrow C-F k-spaces X. (C) 2017 Elsevier B.V. All rights reserved.