A Characterization of the Least Cardinal for Which the Baire Category Theorem Fails

Let k be the least cardinal such that the real Une can be covered by k many nowhere dense sets. We show that k can be characterized as the least cardinal such that "infinitely equal" reals fail to exist for families of cardinality k. Let Baire (k) stand for: "The real line is not the union of k many nowhere dense sets (a set is nowhere dense iff its closure has no interior)". The property was extensively studied in Miller (1981) and Miller (1982). It is easily seen (see Kuratowski (1966)) that: not Baire (k) iff some separable, completely metrizable space is the union of « many nowhere dense sets iff every separable, completely metrizable space without isolated points is the union of k many nowhere dense sets. For example, we may replace the real line by Cantor space (2")), or Baire space (to"). Recall that have the discrete topology, and 2" and w" have the product topology. Let Uniformity (k) stand for the proposition "Every subset of the real line of cardinality less than or equal to k is meager (a set is meager iff it is the union of countably many nowhere dense sets)". Let us recall some standard terminology: | AT | is the cardinality of X, (for any cardinal k) [X]k = {Y] Y C X, ] Y\= «}, [*]""[Y\ Y Q X, \ Y\*¿k}, Ve0« abbreviates "for all but finitely many «", and 3°°n abbreviates "there are infinitely many «". Consider the following properties: Different (k) iff VA G [wf" 3X E [u\u 3/Eio" Vg G A V°°nEXf(n)^g(n); Equal (k) iff VA G [w"]*" VB G [[w]1"]*" 3/G w" Vg G A VXEB3°°nE\X(f(n) = g(n)). Received by the editors April 20, 1981. A MS ( MOS) subject classifications ( 1970). Primary 02K05. 'Research partially supported by an NSF Grant. ©1982 American Mathematical Society 0O02-9939/82/OOOO-O214/$O2.25 498 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use CHARACTERIZATION OF THE LEAST CARDINAL 499 In Miller (1981) it was shown that Uniformity (k) iff Different («). A less satisfactory property was found equivalent to Baire (k). The purpose of this note is to prove Theorem. Baire (k) iff Equal (k). To see that Baire (k) implies Equal (k), note that for any g G co" and X G [w]", {/ G w" IV« G Xf(n) =£■ g(n)} is closed nowhere dense in co". Now let us prove that Equal (k) implies Baire (k). Let Independent (k) stand for: "Vie[[üi]"]*"3Ze [a]aVXEB\Xn Z\ = \X-Z\= oo". Lemma 1. Equal (k) implies Independent (k). Proof. Let B = {Xa ] a < k} and choose X'a G [Xa]u for / = 0,1 so that X° n Xxa = 0. Choose g'a G co" so that g¿(X¡a) = {/}. By Equal (k) let/ G co" be such that for all a and i, 3°°n G X'af(n) = g'a(n) = /. Then Z = /"'{0} does the job. D Definition. Z G [co]" is /-uncrowded iff V«, m G Zn ^ m -» | n — m \ > /. Definition. /-Uncrowded (k) iff V73 G [[co]"]"6" 3Z /-uncrowded VXEB \xnz\= co. Lemma 2. The following are equivalent. (A) Independent (k), (B) 2-Uncrowded(k), (C) for all I < oo I-Uncrowded (k). Proof. Let us first prove that (A) implies (B). I claim there exists T G [co]" such that for every X G B, \ X — T\= oo. To see this, note that Independent (k) implies k < c =| <o" | . A well-known theorem of Sierpinski (1928) says that there exists an almost disjoint family of cardinality c, i.e. there exists Ma G [oo]" for a < c such that for all a ¥= ß, \ Ma n Mß | < co. Since the Ma are almost disjoint and k < c, for some a < c for all X G B ] X — Ma ] = co. Let T be any such Ma. Let E be the even integers and O the odd integers. Without loss of generality we may assume that for all X G B, X Ç E T or X C O T. Let T = {a„: n < co} be an enumeration in increasing order and for any a and b let (a, b) = {n Gco|a<n < b}. For each X G B let X* {n | (a„, an+x) C\ X^ 0}. Let «/ be independent with respect to {X* | X G 5} (i.e. for all A"*, | X* W\ = ] X* n W| = co). Let Z= U {(a„,a„+1)n£|«GlF} U {(a„, a„+1) n O | n £ IF}. It is easily checked that Z is 2-uncrowded and for all X G B, \ Z n A | = co. (B) implies (C) is proved by induction on /. Suppose Z is /-crowded and for all X E B, | A n Z | = co. Let Z = {an: n < oo} (increasing order) and for each X E B, X* = {« | an G X}. Let Q be a 2-uncrowded set such that for all A G B, \X* (1 Q\= oo. Then (an|«Gg}isa 2/-uncrowded set meeting each element of B in an infinite set. Now we prove (B) implies (A). Let {Wa ] a < k} ç [co]". For each a < k, let Wea = {2« | n G Wa) and Wa° = {2« + 1 | « G Wa). Let Z be a 2-uncrowded set such that for each a < k, \ Z n W„e | = | Z n PFá° | = co. Let Q = {n ] 2« G Z}. Then License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use