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 ) . 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 ˜ .