From: "Brian M. Scott" Subject: Re: Definition of a discontinuous function Date: Sun, 13 Jun 1999 00:38:56 -0400 Newsgroups: alt.algebra.help,sci.math Keywords: topologies on power set (Vietoris topology, Pixley-Roy topology) Ronald Bruck wrote: > In article <37626260.B6B5B5F3@hut.fi>, Pertti Lounesto > wrote: > :"Brian M. Scott" wrote: > :> Pertti Lounesto wrote: > :> > And what if you define f: R -> R U P(R), by > :> > > :> > sin(1/x) for x not= 0 > :> > f(x) = > :> > [-1,1] for x = 0? > :> You will be obliged first to define a topology on > :> R U P(R) if you wish to talk about continuity of f. > :That is right. If we simplify things, and consider only > :mappings f:R->P(R), what interesting topologies could > :we use on P(R)? > Ah, missed seeing this before I replied to the earlier post. > On P(R) itself, I dunno of a useful one. If X is a topological space, let A(X) be the collection of non-empty subsets of X topologized as follows. For each n in N and each (n+1)-tuple of open sets in X, define <> = {A in A(X): A is a subset of V(0) U ... U V(n) and A meets each V(i) in a non-empty set}. The sets <> form a base for the Vietoris topology on A(X). Another topology on A(X) is the Pixley-Roy topology; its base is the collection of sets [A,V], for A in A(X) and V open in X, where [A,V] = {S : A is a subset of S and S is a subset of V}. This is finer than the Vietoris topology and is most interesting when X is metrizable. In each case it's often more interesting to look at a subspace, e.g., the finite, compact, or closed non-empty subsets of X instead of all of them. For Pertti's example we could use the compact sets, but it's not continuous wrt either topology: for each n [-1,1] has a Vietoris nbhd in which every point is a set of cardinality at least n. Brian M. Scott