【摘 要】
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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
【机 构】
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Department of Mathematics
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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.
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