A Hilbert cube compactification of a function space from a Peano space into a one-dimensional locally compact absolute retract

摘要

Let X be an infinite Peano space (i.e., locally compact, locally connected, separable metrizable space) and let Y be a 1-dimensional locally compact AR. The space of all continuous functions from X to Y with the compact-open topology is denoted by C(X, Y). In this paper, we show that if X is non-discrete or Y is non-compact, then the function space C(X, Y) has a natural compactification C(X, Y) such that the pair (C(X,Y),C(X,Y)) is homeomorphic to (Q,s), where Q = [-1,1]~N is the Hilbert cube and s = (-1, 1)~N is the pseudo-interior of Q. In fact, the space Y has a dendrite compactification Y such that the remainder Y Y is closed and consisting of end points, and the compactification C(X, Y) is the space of all upper semi-continuous continuum-valued functions from X to Y.