Descriptive locale theory, by John Isbell

Topology Atlas Document # zaaa-43.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 1 (1996), 4.

Descriptive locale theory
by

John Isbell

(State University of New York at Buffalo, New York, U.S.A)

Descriptive locale theory concerns Borel sublocales of the real line R or of other metrizable spaces X, defined like Borel sets but in the lattice Subl(R) of all sublocales of R (or X). It has two and a half results so far. First,

(1): a sublocale of a metrizable space X is complemented in Subl(X) if and only if it is an F_sigma and G_delta. (Every F_sigma in X is a subspace.)

(2): Every absolutely Borel (metrizable) locale is a space.

(3): Maybe every spatial Borel sublocale S of R is SIGMA_3 (i.e. G_delta sigma ) -- it is true if S is PI_4.

For (1), see John Isbell, First steps in descriptive theory of locales, Trans AMS 327 (1991), 353-371; corrections, ibid. 341 (1994) 467-468. For (2) and (3), Till Plewe, Localic products of spaces, Proc. London MS, to appear.

Received by the editors: December 8, 1995.