摘要
Let L be a continuous semilattice. We use USC(X, L) to denote the family of all lower closed sets including X × {0} in the product space X × AL and ↓1 C(X,L) the one of the regions below of all continuous maps from X to AL. USC(X, L) with the Vietoris topology is a topological space and ↓C(X, L) is its subspace. It will be proved that, if X is an infinite locally connected compactum and AL is an AR, then USC(X, L) is homeomorphic to [-1,1]ω. Furthermore, if L is the product of countably many intervals, then ↓ C(X, L) is homotopy dense in USC(X,L), that is, there exists a homotopy h : USC(X,L) × [0,1] →USC(X,L) such that h0 = idUSC(X,L) and ht(USC(X,L)) C↓C(X,L) for any t > 0. But ↓C(X, L) is not completely metrizable.
Let L be a continuous semilattice. We use USC (X, L) to denote the family of all lower closed sets including X×{0} in the product space X×∧Lambda L and ↓C(X,L) the one of the regions below of all continuous maps from X to ∧L. USC}(X, L) with the Vietoris topology is a topological space and ↓C(X,L) is its subspace.It will be proved that, if X is an infinite locally connected compactum and ∧L is an AR, then USC(X,L) is homeomorphic to [-1,1]w. Furthermore, if L is the product of countably many intervals, then↓C(X, L) is homotopy dense in USC (X, L), that is, there exists a homotopy h:USC (X, L)×[0,1]→USC (X, L) such thath_0= id USC (X, L) and ht( USC (X, L)(∪)↓ C (X, L) for any t>0. But↓C}(X, L) is not completely metrizable.
基金
This work was supported by the National Natural Science Foundation of China(Grant No.10471084)
by Guangdong Provincial Natural Science Fundation(Grant No.04010985).
