[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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