Conference rusure::math

1892.0. "The topology of movies" by MOVIES::HANCOCK () Tue Sep 06 1994 17:06

Baire space is the topological space whose points are the one place
functions f : Nat -> Nat, with basic neighbourhoods the sets [c]
of functions with a given finite sequence c : Seq(Nat) as common
prefix.  I know little about it, and have an open-ended question
which, if it isn't too stupid, maybe someone can shed some light on.

(For all you continued fraction fans, it's homeomorphic to the
irrational numbers between 0 and 1 -- the map from Baire space
to the irrationals takes f to the continued fraction:
   1/(1+f(0)+1/(1+f(1)+1/(1+f(2)+...))). ) The Baire topology is
induced by the metric which makes the distance between
2 points f and g the reciprocal of the least i such that f(i) and
g(i) differ.

The set of functions f : Nat -> S, where S is a set of discrete states
for a non-detrministic state machine can be given a topology in the same way.
The points of this space can be thought of as infinite "movies" that might be
taken of the state machine.  For pretty obvious reasons, movies in which
no two adjacent states are the same (except possibly for infinite
repetition of a final state) are particularly interesting.

So what is Baire space like, if you restrict it to functions such that
f(i) and f(i+1) differ, unless f(i), f(i+1), f(i+2), .. are all the
same?  Anything worth writing home about?  How about the irrationals
whose continued fraction expansions contain no adjacent repetitions
(except possibly for infinite final repetition)?  How about the
continuous maps from this space to itself?