Compact subsets of {0,1}^N

On Tue, 28 Jun 2005, David C. Ullrich wrote:
> William Elliot <marsh@xxxxxxxxxxxxxxxxxx> wrote:
> >>
> >> >A closed nonnul subset K without isolated points of S = {0,1}^N
> >> >is homeomorphic to S.
>
> f maps S into K:

This I did by defining f(s) to be the unique, as it turns out, element of
the intersection of a descending sequence of nonnul closed subsets of K.

> f maps S onto K: Let's say that a finite string t in T is an n-sibling
> if t has length n and there exists t' in T such that t' <> t, t' has
> length n, and t and t' agree in the first n-1 places. Say that t in T is
> a sibling if it is the empty string or an n-sibling for some n. By
> construction the range of F is exactly the set of all siblings.
>
This I'm still puzzling. Assume s in K and lets take the first step,
showing F(0) or F(1) is a prefix for s. Would you do a slow motion
replay?

> Now suppose that k is in K. Since K has no isolated
> points there exists a cofinal sequence of initial
> segments of k, each of which is a sibling. Since
> each sibling is in the range of F it follows that
> k is in the range of f.
>
> >> >[stuff I dunno nothing about snipped]
> >Ah shucks. What don't you know about?
>
> Plenty. (Flattery may get you somewhere, but if you want
> me to worry about things that I don't feel like worrying
> about send money.)
>
Is this proof you came up with a proof of your own making?
How do you want payment, in bhats, dongs or pesos? ;-)

let Sf = set of binary sequences in S that eventuate to 0.
Sf is a countable dense subset of S = {0,1}^N.
Let D be a countable dense subset of S. D has no isolated points.
Let { d_j | j in N } be an enumeration of D

Well posh and bother. You'd think there'd be a natural bijection between
Sf and B = set of finite binary sequences. Nothing so simple like just
tack on 000... to every b in B or just clip the training 0's from s in Sf.

So less I worry you more than just your two cents worth, whoops better
make that three cents worth due to inflation, where would Sf apply for
naturalization onto the finite world of B?


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