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

作者：Koshino Katsuhisa^*; Sakai Katsuro

来源：Topology and Its Applications, 2014, 161: 37-57.

DOI：10.1016/j.topol.2013.09.008

• Function space
• Hyperspace
• The compact-open topology
• The Fell topology
• Dendrite
• Hilbert cube compactification

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) over bar (X, Y) such that the pair ((C) over bar (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) over tilde \ 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) over tilde.

• 出版日期2014-1-1