From: "Brian M. Scott" <BMScott@stratos.net>
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
> <Pertti.Lounesto@hut.fi> 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 <V(0), ..., V(n)> of 
open sets in X, define <<V(0), ..., V(n)>> = {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 <<V(0), ..., V(n)>> 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