[FOM] Work on Tarksi Closures?
Rex Butler
rexbutler at hotmail.com
Tue Nov 23 17:21:22 EST 2004
A while back I came across a book at the library called 'Closure Spaces and
Logic' by Martin and Pollard which I found rather interesting.
Unfortunately, it didn't have many references. It begins essentially with a
definition of Tarksi's:
<S,Cl> is a Tarski Space if
1. <S,Cl> is a closure space, i.e. Cl : P(S) -> P(S) is expansive,
increasing, and idempotent.
2. S is countable
3. Cl is finitary, i.e. Cl(A) = U {Cl(A_0) : A_0 is a finite subset of A}
One can then define such things as consisteny, completeness, independence,
etc. along with the standard connectives. Also on defining continuous maps
between closure spaces, one has a category of Tarkski spaces.
Question: How much work has been done on such spaces? I have heard the
Polish mathematicians have done a significant amount, but I have no idea in
what direction. How promising is this direction?
In any case, I have found the idea of a closure operator incredibly
illuminating. One learns soon enough the idea of a quotient construction in
mathematics, but I have never had anyone directly point out the ubiquity of
closure. This seems rather odd to me, considering that behind every
induction and every "the smallest ___ such that ___" lies a closure operator
of some sort. Is there a reason why closures are not given more
foundational weight? An inquiring graduate student wants to know. Thanks.
Rex Butler
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