One has just to notice in the proof of Rudin that the supports supp([h.sub.i]) of the constructed functions [h.sub.i] are compact sets (a closed subset
of a compact set is compact).
If F is a closed subset
of X, then F = ker([M.sub.F]), where [M.sub.F] is as defined in 1.1.
For any closed subset
F of [X.sup.p] such that [DELTA] [subset or equal to] F, there exists a continuous mean-type mapping M : [X.sup.p] [right arrow] [X.sup.p] such that Fix(M) = F.
According to Luzin's Theorem, there is a closed subset
[D.sub.[epsilon]] [subset] [[0, T].sup.N] such that m([D'.sub.[epsilon]]) [less than or equal to] [epsilon], and the functions [mathematical expression not reproducible] are continuous.
Let (X, [[sigma].sub.b]) be a complete b-metric-like space with parameter s [greater than or equal to] 1 and f g be self-mappings on X such that f(X) [subset] g(X), f(X) or g(X) is a closed subset
of X, and a : X x X [right arrow] [0, +[infinity]] a given mapping.
Suppose that X is a compact connected polyhedron without local cutpoints and A is a closed subset
imbedded inside a subpolyhedron K that can be by-passed in X, that is, every path C in X with C(0), C(1) [member of] X - K, is homotopic to a path C' in X - K relative to the endpoints.
The left P'-invariant closed subsets
of G/P are described in the following Hasse diagram.
Since [psi] (Y) [subset] [phi](Y x X) [subset] K [subset] X, where K is a compact set, in view of Lemma 2.2, it is sufficient to show that the graph [[GAMMA].sub.[psi]] of [psi] is a closed subset
of Y x X.
Since, for any particular point topological space X with the particular point p [member of] X, any nonempty open subset includes p, it follows that the only closed subset
that includes p is X.
Let (A, B) be a pair of nonempty closed subset
X of a p-bounded and S-complete Hausdorff uniform space (X, [GAMMA]) such that [A.sub.0], [B.sub.0] [not equal to] [theta] and p is an [J.sub.av]- distance on X.