I. I. Parovičenko

On a Universal Bicompactum of Weight \(\aleph\)

(Presented by Academician P. S. Aleksandrov on 20 XI 1962)

1. According to a well-known theorem of P. Aleksandrov, the Cantor discontinuum \(\Delta_{\aleph_0}\) has the universality property in the class of bicompacta of weight \(\leqslant \aleph_0\), consisting in the fact that every bicompactum of weight \(\leqslant \aleph_0\) is its continuous image. P. Aleksandrov also gave a simple topological definition of \(\Delta_{\aleph_0}\) as a zero-dimensional perfect bicompactum of weight \(\aleph_0\). In \((^1)\) A. Esenin-Volpin, using the generalized continuum hypothesis, proved the existence, in the same sense, of a universal bicompactum for any weight \(\mathfrak m\).

In the present paper it will be proved (using the continuum hypothesis, which below is always assumed unless the contrary is stipulated) that the Čech remainder on the natural number series
\[ \Delta_{\aleph}=\beta N\setminus N \]
is a universal bicompactum for the class of bicompacta of weight \(\leqslant \aleph\), and a topological definition of \(\Delta_{\aleph}\) will also be given and a theorem for \(\Delta_{\aleph}\) obtained that is completely analogous to the theorem for \(\Delta_{\aleph_0}\).

1. Below we use the following notation in the Boolean algebra \(L\): \(a+b\), \(ab\), \(b'\), \(ab'=a\setminus b=a-b\) (the latter when \(b\leqslant a\)); the least element is denoted by \(0\); the principal ideal \(\{x\mid x\leqslant a\}\) is denoted by \(L_a\), the dual principal ideal \(\{x\mid x\geqslant a\}\) by \(L^a\). As Nowak showed \((^2)\), the Boolean algebra of all open-closed sets of \(\Delta_{\aleph}\) is isomorphic to the Boolean algebra \(L_{\aleph}\), defined as follows.

As elements of \(L_{\aleph}\) one takes equivalence classes in the system of all subsets of the natural number series, where \(E\sim G\) if \((E\setminus G)\cup(G\setminus E)\) is finite, and for \(e,g\in L_{\aleph}\) one has \(e<g\) if \(G\setminus E\) is infinite and \(E\setminus G\) is finite, \(E\in e\), \(G\in g\).

We shall say that in a partially ordered set \(T\) the following hold: the simplest separability, if for any \(e<h\) from \(T\) there exists \(g\) such that \(e<g<h\); Cantor separability, if for any set
\[ e_1<\cdots<e_n<\cdots<h \]
of type \(\omega+1\) there exists \(g\) such that \(e_n<g<h\); and du Bois-Reymond separability, if for any set
\[ e_1<\cdots<e_n<\cdots<h_n<\cdots<h_1 \]
of type \(\omega+\omega^*\) there exists \(g\) such that \(e_n<g<h_n\). We shall say that a zero-dimensional bicompactum has the corresponding separability if the Boolean algebra of its open-closed sets has it. In particular, the simplest separability is equivalent to the requirement that the bicompactum be perfect. The remainder \(\Delta_{\aleph}\), besides the simplest separability, also has the Cantor and du Bois-Reymond separabilities, as is testified (in view of Nowak’s isomorphism) by the theorems of the same name on subsets of the natural numbers \((^3\), p. 715). However, for zero-dimensional bicompacta of weight \(\aleph\), in general, none of the three separabilities implies another, and no two imply the third. Indeed, from Cantor separability it follows that the intersection of a strictly decreasing sequence of open-closed sets contains an interior point, but the bicompactum
\[ \Bigl[\bigcup_n e_n\Bigr](\Delta_{\aleph})\quad (e_1\subset e_2\subset\cdots \text{ and open-closed in } \Delta_{\aleph}) \]
does not have this property, although the other two separabilities hold for it.

From du Bois-Reymond separability it follows that in the bicompactum there are no cappa-

points ((4), p. 912), while the ordered bicompactum of type \(2^{\omega_1^*}\) (notations from (5)) contains them, although it satisfies the other two separation axioms.

Theorem 1. Every perfect zero-dimensional bicompactum of weight \(\aleph\) with the separation axioms of Cantor and of Du Bois-Reymond is homeomorphic to \(\Delta_{\aleph}\), and, independently of the continuum hypothesis, \(\Delta_{\aleph}\) is continuously mapped onto any bicompactum of weight \(\leq \aleph_1\).

Lemma 1. In a Boolean algebra \(L\) with Cantor’s separation axiom, every ideal cofinal with no more than a countable set is closed (see (6), p. 95).

Let \(A\) be an ideal in \(L\), cofinal with the sequence
\[ a_1<\cdots<a_n<\cdots . \]
In what follows we shall regard \(L\) as the family of all open-closed sets of some zero-dimensional bicompactum \(\Delta\), while preserving, however, the structural notation. Let \(a\leq b\) for every \(b\supset \bigcup_n a_n\) and a fixed \(a>0\), and moreover for all \(a_n\) let \(a\not\leq a_n\). The intersection of all the indicated \(b\)’s is
\[ \left[\bigcup_n a_n\right](\Delta). \]
Consequently,
\[ a\subseteq\left[\bigcup_n a_n\right], \]
and since \(a\) is open, we have
\[ a\cap\left(\bigcup_n a_n\right)=\Lambda \]
and \(aa_n>0\) for \(n\geq n_0\). But, by assumption, \(a\setminus a_i>0\) and \(a\setminus a_n\) decrease. If \(\{a\setminus a_n\}\) does not stabilize, then, by Cantor’s separation axiom, there exists a \(c\) such that
\[ 0<c<a\setminus a_n; \]
hence
\[ 0<c\leq a_n\subseteq\left[\bigcup_n a_n\right] \]
and
\[ c\cap\left(\bigcup_n a_n\right)=\Lambda, \]
which is impossible. The stabilization case does not require use of the separation axiom.

Lemma 2. If in a Boolean algebra \(L\) with the separation axioms of Cantor and of Du Bois-Reymond an ideal \(A\) is cofinal with no more than a countable set and \(C=\{c_i\}\) is a set of no more than countably many elements in \(L\setminus A\), then there exists a principal ideal \(L_g\) such that \(A\subseteq L_g\) and \(C\subseteq L\setminus L_g\).

Let \(A\) be cofinal with
\[ a_1<\cdots<a_n<\cdots . \]
By Lemma 1, for each \(c_i\) choose \(e_i\) such that \(A\leq e_i\) and \(c_i\not\leq e_i\). Put
\[ g_j=\bigwedge_{i=1}^{j} e_i; \]
then
\[ g_1\geq\cdots\geq g_j\geq\cdots, \]
with
\[ g_j\leq e_i \quad (i=1,\ldots,j), \]
so that
\[ c_i\not\leq g_j \quad (i=1,\ldots,j). \]
If \(\{g_j\}\) stabilizes at \(g_{j_0}\), then \(L_{g_{j_0}}\) is the required ideal. If, however, \(\{g_j\}\) does not stabilize, then, by the Du Bois-Reymond separation axiom, there exists a \(g\) such that
\[ a_n<g<g_j \]
and
\[ c_i\not\leq g; \]
then \(L_g\) is the required ideal.

Lemma 3. Let in a Boolean algebra \(L\) with the separation axioms of the simplest kind, of Cantor, and of Du Bois-Reymond there be given three no more than countable sets
\[ A=\{a_l\},\quad B=\{b_m\},\quad C=\{c_n\}, \]
with
\[ a_1<\cdots<a_l<\cdots<b_m<\cdots<b_1, \]
and suppose that for any \(l,m,n\) one has \(c_n\not\leq a_l\) and \(b_m\not\leq c_n\). Then there exists a \(d\) such that
\[ a_l<d<b_m \]
and \(d\) is not comparable with any \(c_n\).

Let
\[ \widetilde A=\bigcup_m\{x\mid x\leq a_l\},\qquad \widetilde B=\bigcup_l\{x\mid x\geq b_m\}; \]
then \(\widetilde A\) is an ideal (\(\widetilde B\) is a dual ideal), cofinal (coinitial) with no more than a countable set, and
\[ \widetilde A<\widetilde B. \]
Using the separation axioms of Cantor and Du Bois-Reymond, we find such \(g_0\) and \(g_1\) that
\[ \widetilde A\leq g_0<g_1\leq\widetilde B, \]
and, by Lemma 2 and the dual proposition to it, choose such \(h_0\) and \(h_1\) that
\[ \widetilde A\subseteq L_{h_1},\qquad \widetilde B\subseteq L^{h_1},\qquad C\subseteq L\setminus (L_{h_0}\cup L^{h_1}). \]
Let
\[ t_0=g_0h_0,\qquad t_1=g_1+h_1; \]
then
\[ \widetilde A\leq t_0<t_1\leq\widetilde B,\qquad C\subseteq L\setminus (L_{t_0}\cup L^{t_1}). \]
Put
\[ c_n^0=c_nt_1; \]
since
\[ c_n\in L^{t_1}, \]
we have
\[ c_n^0<t_1. \]
Let
\[ e_p=\bigvee_{n=1}^{p} c_n^0, \]
and put
\[ q_1=e_1, \]
\[ q_2=q_1+(e_2-q_1),\ldots,\qquad q_p=q_{p-1}+(e_p-q_{p-1}),\ldots,\qquad r_0=t_0,\ldots, \]
\[ \ldots,\ r_p=t_0+q_p,\ldots,\qquad t_0=r_0\leq r_1\leq r_2\leq\cdots<t_1. \]
Using the separation axiom of the simplest kind or Cantor’s axiom, we find an \(r\) such that
\[ r_p<r<t_1 \quad (p=0,1,\ldots), \]
and then
\[ d=t_0+(t_1-r) \]
is the required element.

Now Theorem 1 is proved analogously to Rudin’s theorem in (7), 4.7, on homeomorphisms of \(\Delta_{\aleph}\) onto itself, but as applied to two Boolean algebras and with replacement of Lemma 4.8 from (7) by our Lemma 3 (cf. also (1)).

Corollary. The class of bicompacta of weight \(\leq \aleph\) coincides with the class of all bicompact extensions of the natural row.

2. Let \(C_{\aleph}=\{y\}\) be the lexicographically ordered set of all sequences of type \(\omega_1\) of real numbers \(y_\xi\), \(0\leq y_\xi\leq 1\), and let \(I_{\aleph}\) be the ordered set obtained after removing from \(C_{\aleph}\) all kappa-points.

Theorem 2 is analogous to the theorem on the universality of the Baire \(0\)-space.

Theorem 2. The ordered space \(I_{\aleph}\) is condensed onto \(\Delta_{\aleph}\) and therefore is mapped continuously onto any bicompactum of weight \(\leq \aleph\).

Since \(\Delta_{\aleph}\) is a condensation of \(T^1\Delta_{\aleph}\) \((^8)\), it is enough for us to prove that \(I_{\aleph}\) and \(T^1\Delta_{\aleph}\) are homeomorphic. Let \(\mathfrak{G}=\{\Gamma_\nu\}_{\nu<\omega_1}\) be an enumerated collection of all nonempty open-closed sets of \(\Delta_{\aleph}\); here we assume that, if \(\zeta\) is zero or limit, then \(\Gamma_{\zeta+2k}\) and \(\Gamma_{\zeta+2k+1}\) are complements in \(\Delta_{\aleph}\) \((0\leq k<\omega)\). Let \(x=\{i_\xi\}_{\xi<\omega_1}\) be a sequence of 0’s and 1’s.

Define \(\mathfrak{D}(x)=\{D_\xi(x)\}\) by induction: \(D_0(x)=\Gamma_0\) for \(i_0=0\), and \(D_0(x)=\Gamma_1\) for \(i_0=1\); if \(D_\xi(x)\) are defined for \(\xi<\eta\), then for limit \(\eta\)

\[ D_\eta(x)=\bigcap_{\xi<\eta}D_\xi(x), \]

and for \(\eta=\eta_0+1\),

\[ D_\eta(x)=D_{\eta_0}(x)\cap\Gamma_{\nu_0}, \]

if \(i_\eta=0\), and

\[ D_\eta(x)=D_{\eta_0}(x)\cap\Gamma_{\nu_0+1}, \]

if \(i_\eta=1\), where \(\nu_0=\nu_0(\eta)\) is the least of those \(\nu\) for which simultaneously

\[ D_{\eta_0}\cap\Gamma_\nu\supset\Lambda,\qquad D_{\eta_0}\cap\Gamma_{\nu+1}\supset\Lambda. \]

It is clear that \(\nu_0(\eta)\) strictly increases. The sequence \(\mathfrak{D}(x)\) decreases and has a nonempty intersection by virtue of the bicompactness of \(\Delta_{\aleph}\), and from the construction it follows easily that this intersection contains a single point; and hence we shall write \(\Delta_{\aleph}=\{x\}\). Let us prove that \(\mathfrak{D}=\bigcup_x\mathfrak{D}(x)\) forms an open base in \(T^1\Delta_{\aleph}\).

Indeed, let

\[ Ox=\bigcap_{n<\omega}G_n(x), \]

where \(G_n(x)\) are neighborhoods of \(x\) in \(\Delta_{\aleph}\). Since

\[ \bigcap_n D_{\xi_n}(x)=x\in G_n(x), \]

then, by the bicompactness of \(\Delta_{\aleph}\), there exists \(\xi_n\) such that

\[ D_{\xi_n}(x)\subseteq\bigcap G_n(x); \]

taking \(\xi'>\xi_n\), we have

\[ D_{\xi'}(x)\subseteq\bigcap_n D_{\xi_n}(x)\subseteq\bigcap_n G_n(x)=Ox. \]

Now enumerate all even limit numbers: \(\tau_0=\omega,\ \tau_1=\omega2,\ldots,\tau_\pi,\ldots\), and let

\[ \mathfrak{D}_\pi=\{D_{\tau_\pi}(x)\mid x\in\Delta_{\aleph}\}; \]

then

\[ \mathfrak{D}=\bigcup_{\pi<\omega_1}\mathfrak{D}_\pi, \]

where \(\{\mathfrak{D}_\pi\}\) is a sequence of disjoint open-closed covers of \(T^1\Delta_{\aleph}\), and each element of \(\mathfrak{D}_\pi\) contains \(\aleph\) elements of \(\mathfrak{D}_{\pi+1}\). Let \(C(y_0,\ldots,y_\xi,\ldots)\) be the set of all points of \(C_{\aleph}\) beginning with the complex

\[ (y_0,\ldots,y_\xi,\ldots)_{\xi<\eta<\omega_1}, \]

and let

\[ \mathfrak{C}_\eta=\{C(y_0,\ldots,y_\xi,\ldots)\}; \]

then \(\{\mathfrak{C}_\eta\}_{\eta<\omega_1}\) is a sequence of covers of \(C_{\aleph}\) by disjoint systems of segments, while the order type of \(\mathfrak{C}_\eta\) is equal to \(0^{\eta^*}\), and therefore the kappa-points of \(C_{\aleph}\), and only they, are the ends of segments from

\[ \bigcup_{\eta<\omega_1}\mathfrak{C}_\eta. \]

Removing them, we obtain a sequence of interval covers of \(I_{\aleph}\), and the joining of the latter will give a base of \(I_{\aleph}\), isomorphic to \(\mathfrak{D}\), whence the required homeomorphism follows.

We shall say that \(M\) is of the \(\aleph\)-category in \(S\) if \(M\) is (not) representable as the union of \(\leq\aleph\) nowhere dense sets in \(S\). It is easy to see that \(\Delta_{\aleph}\) is of the \(\aleph\)-category, and if \(\Pi\) is the set of all \(P\)-points of \(\Delta_{\aleph}\) \((^7)\), then \(\Delta_{\aleph}\setminus\Pi\) is of the \(\aleph\)-category, since \(\Delta_{\aleph}\setminus\Pi\) is the union of the boundaries of a system of \(\aleph^{\aleph_0}=\aleph\) all countable intersections of open-closed sets of \(\Delta_{\aleph}\). By modifying the proof of Theorem 2 one obtains

Theorem 3. The set of all \(P\)-points of \(\Delta_{\aleph}\) is homeomorphic to \(I_{\aleph}\), and therefore every ordered space of cardinality up to the set \(I\) is of the \(\aleph\)-category.

3. Theorem 4. All maximal ordered subsets of \(L_{\aleph}\) are similar and have type \(1+\eta_1+1\), where \(\eta_1\) is the normal Hausdorff type \(((^9)\), p. 181).

This theorem follows at once from the separability of \(L_{\aleph}\) and Theorem II of \((^9)\), p. 181. Theorem 4 gives a concrete embodiment of several somewhat indefinite thoughts of Luzin on the “Pythagoras phenomenon” on the “transfinite line” \((^3)\), p. 721. In particular, since the Dedekind completion of a set of type \(1+\eta_1+1\) is like \(C_{\aleph}\) (this also follows easily from the cited theorem of Hausdorff), and \(C_{\aleph}\) has cardinality \(2^{\aleph}\), there turn out to be more of Luzin’s “transfinite irrationalities,” as of the irrationalities of the ordinary line, than of “real” (rational) points: they correspond to Dedekind cuts in a set of type \(1+\eta_1+1\).

At the same time one also obtains an immediate solution of all Luzin’s problems from \((^3)\), p. 721, which, however, is not new \((^2)\).

Kishinev State
University

Received
16 XI 1962

CITED LITERATURE

\(^1\) A. Esenin-Volpin, DAN, 68, No. 4 (1949).
\(^2\) I. Novak, Czechoslovak Math. J., 3 (78), issue 4, 385 (1953).
\(^3\) N. Luzin, Collected Works, 2, 1958.
\(^4\) P. S. Urysohn, Works on topology and other fields of mathematics, 2, Moscow, 1951.
\(^5\) F. Hausdorff, Set Theory, Moscow—Leningrad, 1937.
\(^6\) G. Birkhoff, Lattice Theory, Moscow, 1952.
\(^7\) W. Rudin, Duke Math. J., 23, No. 3 (1956).
\(^8\) I. I. Parovichenko, DAN, 115, No. 5 (1957).
\(^9\) F. Hausdorff, Grundzüge der Mengenlehre, N. Y., 1949.