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Cluster sets of nets

1969; American Mathematical Society; Volume: 22; Issue: 2 Linguagem: Inglês

10.1090/s0002-9939-1969-0276922-4

1088-6826

B. J. Pettis,

Mathematics and Applications

We give here a few simple results on cluster sets of nets in topological spaces with some applications to real variables and elementary topological algebra. If ,u is a net in topological X and lim sup ,u denotes the set of its cluster points, then f(lim sup ,u) Clim sup f(,u) for any continuous f on X; the inclusion may be proper. Theorem A asserts that the two sets are equal if every subnet of ,u clusters and if the range of f is Hausdorff. This fact, curiously enough, gives deeper insight into certain elementary situations in real variables and is a useful device in topological algebra, as shown in the discussion after Corollary 2. Theorem D concerns nets in uniform spaces and may be considered to be the elementary principle behind Duhamel theorems. This theorem and Corollaries 1 and 2 (known to the author since 1953) were referred to in an abstract [4]. A net in X is any function having a directed set for its domain and for its values nonvoid subsets of the set X. A net ,u is finally in G if {d: pu(d)CG} is a final of the domain of ,u (final segment here meaning a subset containing all the successors of some element), and repeatedly intersects G if {d: u(d)6G 0 } is cofinal. Given two nets ,u and v in X, we call v a subnet of ,u if for each final I of the domain of ,u there is a final J of the domain of v such that for each e in J there is some d in I such that v(e) C,u(d). A basic fact is that any net has a universal point subnet, i.e., a subnet every value of which is a singleton set and which, for each set G in X, is either finally in G or finally in the complement X\G. When X is topological we define a(ju) to be { F: FCX, F closed, ju is finally in F} , and lim sup ,u to be n { F: FCa(ju) }. Equivalently, x is in lim sup u iff every neighborhood of x is repeatedly intersected by ,u and iff x is an adherent point of a(,u). If ,u finally lies in each neighborhood of x, ju converges to x. Some facts: (i) if v is a subnet of ,u then V(,u)Ca(v) and lim sup vClim sup ,u; (ii) if ,u is finally compact, meaning that a(ju) has a compact element, then lim sup /L $0; (iii) if f on X is continuous then f(lim sup ,u) Clim sup f(u) for every net ,u in X; (iv) xElim sup ,u iff some universal point subnet of jt converges to x; (v) defining uXv for any nets u and v to be (,uXv) * (d, e) =,u(d) Xv(e), we have lim sup(,uXv) = (lim sup u) X (lim sup v).