Sequential spaces and functionals on Boolean algebras
by
Bohuslav Balcar
Institute of Mathematics of the Academy of Sciences of the Czech Republic

From history, in year 1962 M. Venkataraman posed the problem to characterise the class of topological spaces which can be specified completely by the knowledge of their convergent sequences.

Clearly such class has to contain all first-countable spaces. In a year 1965 this problem is settled by S. P. Franklin in a very beautiful article [2], where he pointed out the following

(a) A point lies in the closure of a set iff there is a sequence in the set converging to the point. (Fréchet spaces)
(b) A set is open iff every sequence converging to a point in the set is, itself, eventually in the set. (sequential spaces)

All three classes are closely connected with metric spaces, for example:

FACT. Sequential space are exactly quotients of a metric spaces.

FACT. Firs-countable space are exactly continuous images of metric spaces.

We are interested in sequential topologies on Boolean algebras. Investigation of Boolean algebras from the topological point of view leads to uncovering some interesting properties of forcing extensions. For example:

THEOREM. Complete atomless Boolean algebra B adds independent real if and only if there is a sequence x ∈ B^w such that for each subsequence y

limsup x = limsup y   >   liminf y = liminf x.

This approach leads to investigation of functionals on Boolean algebras and especially Maharam algebras in an optimal case.

References

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B. Balcar, T. Jech Weak distributivity, a problem of von Neumann and the mystery of measurability. Bull. Symbolic Logic 12:2, 12, 241-266, 2006

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S. P. Franklin Spaces in which sequences suffice. Fund.Math. LVII: 107-115, 1965.

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T. Pazák Exhaustive Structures on Boolean Algebras. Ph.D. Thesis, (unpublished), 2007.

Date received: July 1, 2008