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

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)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)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)(Q,s), where Q=[−1,1]NQ=[−1,1]N is the Hilbert cube and s=(−1,1)Ns=(−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˜.