Corrections to "First Steps in Descriptive Theory of Locales"

Abstract

The paper [I2] contains one false result, 2.6, and two others whose proofs require substantial repair: 2.5 and 2.24. All errors were discovered by one critical reader, Till Plewe. In 2.5 of [I2], which characterizes those sober spaces X that have a largest pointless sublocale pl(X), the last six words of proof are not true, e.g. in the real line with a generic point adjoined. Argue instead: The meet of {x}~ and pl+(^0 is dense in the irreducible space {x}~ . Now every dense sublocale of an irreducible space Y contains (i.e. D(Y) contains) the generic point y. For every sublocale of any locale is an intersection of complemented sublocales C whose complements are open n closed [I2], so it suffices to show that every such C dense in Y has y in it. Otherwise the complement C would contain y, so no closed subspace except Y contains C , so C is open; and as C is dense, C = 0, contradicting y e C . 2.6 says that for a dense-in-itself regular space X the locale pl(X) has the same weight as X. This is false—refuted by many pairs (X, X') of regular spaces with the same pi (pl(A^) « pl(X')) but different weights, e.g. the space Q of rationals and a subspace of sQ consisting of Q and one more point. The last result in the paper, 2.24, is that (with everything metrizable; in contrast to CVs) no nonzero pointless-absolute Fa locale exists. The correct proof, now presented, amounts to showing that ( 1 ) any nonzero Fa sublocale A of a pointless-absolute Os has a closed sublocale B = pl(C), C a Cantor set; that (2) B is not pointless-absolute Fa , being not Fa in a suitable metrizable extension pl(E) ; and (3) boosting the extension E of B to an extension of A . In [I2], (1) is done correctly in three lines, and four more lines of the proof (the fourth and the last three) do (2). For (3), a pushout construction is proposed; but it is not hard to check that the pushout need not be first countable. Instead use F. Hausdorffs theorem [H]: