UDC 519.50 + 50.01

MATHEMATICS

I. I. PAROVICHENKO

THE BRANCHING HYPOTHESIS AND THE RELATION BETWEEN THE LOCAL WEIGHT AND THE CARDINALITY OF TOPOLOGICAL SPACES *

(Presented by Academician P. S. Aleksandrov on 25 VI 1966)

1. A topological space is called a \(T^\alpha\)-space if in it the intersection of any system of cardinality \(<\aleph_\alpha\) of open sets is an open set. If \(X\) is a topological space, then \(T^\alpha X\) denotes the space on the set \(X\) whose open base is the totality of all possible intersections of systems of cardinality \(<\aleph_\alpha\) of open sets of the space \(X\). The \(T^\alpha\)-product of spaces \(\{X_\lambda\mid \lambda \in L\}\) is the space on the abstract product \(\prod_\lambda X_\lambda\), whose open base is given by fixing any system of indices \(\lambda\) of cardinality \(<\aleph_\alpha\) together with the choice of open sets in the corresponding spaces (the Tikhonov product is a special case of ours when \(\alpha=0\)). A topological space is called \(\aleph_\alpha\)-bicompact if from every one of its open coverings one can extract a subcovering of cardinality \(<\aleph_\alpha\) (cf. \((1)\)). A system of sets is called \(\aleph_\alpha\)-centered if every one of its subsystems of cardinality \(<\aleph_\alpha\) has nonempty intersection. A partially ordered set \(S\) is called a branching system if every one of its initial segments \(S^b=\{x\mid x\in S,\ x<b\}\) is a well-ordered set; here the type of \(S^b\) is called the order of the element \(b\), and the least of the ordinal numbers that is greater than the orders of all elements of \(S\) is called the order of the branching system \(S\). The cardinal number \(\aleph_\sigma\) is called (strongly) inaccessible if it is regular and from \(\mathfrak n<\aleph_\sigma\) it follows that \(2^{\mathfrak n}<\aleph_\sigma\). \(\aleph_0\) is inaccessible; however, in what follows we shall assume \(\aleph_\sigma\) to be uncountable and shall reserve the indicated notation for it.

In \((2,3)\) we proved, for an inaccessible \(\aleph_\sigma\), the equivalence of the following properties:

\((\alpha)\). Every branching system \(S\) of order \(\omega_\sigma\), in which the set of elements of fixed order has cardinality \(<\aleph_\sigma\), contains a well-ordered subset of type \(\omega_\sigma\).

\((\beta)\). Every linearly ordered set of cardinality \(\aleph_\sigma\) contains a subset of at least one of the types \(\omega_\sigma\) or \(\omega_\sigma^*\).

\((\widetilde{\gamma})\). The \(T^\sigma\)-product of \(\aleph_\sigma\) copies of the simple two-point space is \(\aleph_\sigma\)-bicompact.

Remark. Unfortunately, the authors of papers \((4\text{–}8)\) were unaware of our papers \((2,3)\) (see, in particular, our abstract \((9)\)), as a result of which the following were proved again: in \((4)\), the implication \((\alpha)\to(\widetilde{\gamma})\); in \((5)\), \((\widetilde{\gamma})\to(\alpha)\) (Theorem 2.1); in \((6,7)\), \((\alpha)\leftrightarrow(\widetilde{\beta})\) (Theorem 5 and Theorem 1 respectively). See also Chapter 4 of \((8)\).

2. It is easy to see, passing to the Dedekind completion, that \((\widetilde{\beta})\) is equivalent to the condition

* Proof correction note. The content of the paper was reported at the International Congress of Mathematicians on 25 VIII 1966.

\((\beta)\). Every linearly ordered bicompactum whose weight at all points is \(< \aleph_\sigma\) has cardinality less than \(\aleph_\sigma\).

Using the associativity of the \(T^\sigma\)-product, condition \((\widetilde{\gamma})\) can be replaced by the equivalent condition \((\gamma)\):

\((\gamma)\). The \(T^\sigma\)-product of \(\aleph_\sigma\) copies of discrete spaces of cardinality \(< \aleph_\sigma\) is an \(\aleph_\sigma\)-bicompact space.

In \((^3)\), along with the two-point space, the segment \([0,1]\) of the number line with the discrete topology was considered. It turns out (and this constitutes the main aim of the present paper) that one obtains a statement equivalent to \((\beta)\) if in condition \((\beta)\) the requirement of bicompactness is strongly weakened, and the requirement of orderability is altogether discarded; namely, \((\beta)\) (and consequently also \((\alpha)\) and \((\gamma)\)) is equivalent to the following assertion:

\((\beta^+)\). Every \(\aleph_\sigma\)-bicompact Hausdorff space (in particular, every bicompactum) whose weight at all points is \(< \aleph_\sigma\) has cardinality \(< \aleph_\sigma\).

This, in particular, is of interest in connection with the well-known unsolved problem of P. S. Aleksandrov on the cardinality of bicompacta with the first axiom of countability (see \((^{10})\), p. 853, P. S. Aleksandrov’s note No. 6). In fact, the following is obtained.

Corollary. The existence of bicompacta with the first axiom of countability and of uncountable cardinality is incompatible with the existence of strongly inaccessible numbers with property \((\alpha)\).

3. Theorem 1. If \(\aleph_\sigma\) has property \((\alpha)\) and the space \(X\) is an \(\aleph_\sigma\)-bicompact space of weight \(\aleph_\sigma\), then \(T^\sigma X\) is also an \(\aleph_\sigma\)-bicompact space of weight \(\aleph_\sigma\).

Proof. Let \(g=\{G_\lambda\}\) be an open base of \(X\) of cardinality \(\aleph_\sigma\). Then \(h=\{H_\mu\}\), consisting of all possible intersections of subsystems of \(h\) of cardinality \(<\aleph_\sigma\), forms an open base of \(T^\sigma X\) of cardinality
\[ \sum_{\alpha<\sigma}\aleph_\sigma^{\aleph_\alpha}=\aleph_\sigma \]
(see, for example, \((^{11})\), p. 235, 22, b). It is therefore enough for us to show that \(T^\sigma X\) is \([\aleph_\sigma,\aleph_\sigma]\)-compact, i.e., that every \(\aleph_\sigma\)-centered system of closed sets of cardinality \(\aleph_\sigma\) of the space \(T^\sigma X\) has a nonempty intersection. Let such a system be \(\{\Psi_\nu\}\); then each \(\Psi_\nu\) is the intersection of some system \(\{\Phi_{\mu_\nu}\}\) of cardinality \(\leq \aleph_\sigma\) of closed sets \(\Phi_{\mu_\nu}=CH_{\mu_\nu}\), and, by the definition of \(H_\mu\), each \(\Phi_{\mu_\nu}\) is the union of a system of cardinality \(<\aleph_\sigma\) of closed sets \(F_\lambda=CG_\lambda\) of the space \(X\). It is clear that the system of all \(\Phi_{\mu_\nu}=\{\Phi_\tau\}\) is again an \(\aleph_\sigma\)-centered system of closed sets of the space \(T^\sigma X\) of cardinality \(\aleph_\sigma\), which, by associativity of the intersection operation, has the same intersection as the system \(\{\Psi_\nu\}\), and it is enough for us to show that
\[ \bigcap_\tau \Phi_\tau \supset \Lambda . \]
Suppose that \(\tau\) runs through all ordinals \(<\omega_\sigma\) and
\[ \bigcup_{\lambda_\tau} F_{\lambda_\tau}=\Phi_\tau, \]
where, for fixed \(\tau\), \(\lambda_\tau\) runs through a set of cardinality \(<\aleph_\sigma\). Consider the collection \(S\) of all complexes
\[ f^\eta=\{F^0,F^1,\ldots,F^\tau,\ldots \mid \tau<\eta<\omega_\sigma\}, \]
where \(F^\tau=F_{\lambda_\tau}\) and
\[ \bigcap_{\tau<\eta} F^\tau \supset \Lambda; \]
we partially order these complexes by the rule
\[ f^{\eta_1}<f^{\eta_2} \]
if \(\eta_1<\eta_2\) and \(f^{\eta_2}\) extends \(f^{\eta_1}\). Since for fixed \(\tau\) the number of all \(F_{\lambda_\tau}\) is less than \(\aleph_\sigma\), the number of complexes of fixed type is less than \(\aleph_\sigma\). Since \(\{\Phi_\tau\}\) is \(\aleph_\sigma\)-centered, for any \(\eta<\omega_\sigma\) we have
\[ \bigcap_{\tau<\eta}\Phi_\tau \supset \Lambda, \]
whence follows the existence of a complex
\[ \{F^\tau\mid \tau<\eta<\omega_\sigma\} \]
of any type \(\eta<\omega_\sigma\). Therefore our system \(S\) is a branching system of order \(\omega_\sigma\) under the conditions \((\alpha)\), and, consequently, there exists a sequence
\[ \{F^\tau\mid \tau<\omega_\sigma\}, \]
every initial segment of which
\[ \{F^\tau\mid \tau<\eta<\omega_\sigma\} \]
belongs to \(S\) and therefore has a nonempty intersection. It follows that the system
\[ \{F^\tau\mid \tau<\omega_\sigma\} \]
is \(\aleph_\sigma\)-centered and consists of closed-

... sets of the \(\aleph_\sigma\)-bicompact space \(X\), whence

\[ \bigcap_{\tau<\omega_\sigma}\Phi_\tau \supset \times \bigcap_{\tau<\omega_\sigma} F^\tau \supset \Lambda, \]

which was required to be proved.

Remark. Theorem 1 implies \((\tilde{\gamma})\), since the generalized Cantor discontinuum \(D_\sigma\) is the bicompactum of weight \(\aleph_\sigma\), and \(T^\sigma D_\sigma\) is the \(T^\sigma\)-product of \(\aleph_\sigma\) simple doublets. Since \((\gamma)\to(\alpha)\), the assertion of Theorem 1 is even equivalent to \((\alpha)\).

Theorem 2. If a Hausdorff space \(X\) has weight \(<\aleph_\sigma\) at all points and contains everywhere a dense subset \(X_0\) of cardinality \(\aleph_\sigma\), then the cardinality and the integral weight of \(X\) are equal to \(\aleph_\sigma\).

Proof. Let \(f(m)\) be a net in \(X_0\), defined on the directed quasiordered set \(M=\{m\}\) (cf. \((12)\)); by the cardinality of a net we shall mean \(\operatorname{card} M\). For nets \(f(m)\) and \(g(n)\), \(n\in N\), of the same cardinality in \(X_0\), introduce an equivalence relation by the rule: \(f(m)\sim g(n)\) if there exists a similarity mapping \(\varphi:M\) onto \(N\) such that \(g(\varphi(m))=f(m)\). For a fixed \(\aleph_\alpha<\aleph_\sigma\) consider the class of all nets in \(X_0\) of cardinality \(\aleph_\alpha\). Since the largest number of pairwise similar directed quasiorders on sets of cardinality \(\aleph_\alpha\) is \(2\aleph_\alpha\), the number of all equivalence classes for nets in \(X_0\) of cardinality \(\aleph_\alpha\) will be \(\aleph_\sigma \aleph_\alpha 2\aleph_\alpha=\aleph_\sigma\). Now assign to each point \(x\), whose weight is \(\aleph_\alpha\), the class of nets in \(X\) equivalent to the net \(f^x(U)\), \(f^x(U)\in U(x)\cap X_0\), where \(\{U(x)\}=\mathfrak U(x)\) is a fixed base of the space \(X\) at \(x\) of cardinality \(\aleph_\alpha\), ordered by the relation \(\supset\). Obviously the net \(f^x(U)\) converges to \(x\). Since \(X\) is a Hausdorff space, different points correspond to different equivalence classes, for otherwise one net would converge to different points, which is impossible \(((12), p. 67)\). Thus the number of all points of \(X\) whose weight is \(\aleph_\alpha\) is not greater than \(\aleph_\sigma\), and consequently all limit points of \(X\) will be not more than \(\aleph_\sigma \operatorname{card}\sigma=\aleph_\sigma\). Since all isolated points of \(X\) lie in \(X_0\) of cardinality \(\aleph_\sigma\), it follows that \(\operatorname{card}X=\aleph_\sigma\). Combining the bases \(\mathfrak U(x)\) of all points \(X\), we obtain an integral base of cardinality \(\leq \aleph_\sigma\), so that it remains only to prove that the weight of \(X\) cannot be \(<\aleph_\sigma\). But it is easy to see that every \(T_1\)-space of integral weight \(\aleph_\alpha<\aleph_\sigma\) has cardinality \(<\aleph_\sigma\), since in it distinct points correspond to distinct subfamilies of the integral base consisting of all basic neighborhoods of these points, and the cardinality of the space is \(\leq 2\aleph_\alpha<\aleph_\sigma\).

We now prove our implication \((\alpha)\to(\beta^+)\), from which will follow the equivalence of \((\beta^+)\) with the properties \((\alpha)\), \((\beta)\), and \((\gamma)\), since \((\beta^+)\to(\beta)\) is obvious. Suppose the contrary, i.e., that \((\alpha)\) holds, while \((\beta^+)\) does not. Then, in particular, there exists an \(\aleph_\sigma\)-bicompact \(T_2\)-space \(Y\), whose weight at all points is \(<\aleph_\sigma\) and \(\operatorname{card}Y\geq \aleph_\sigma\). Take a subset \(X_0\) of \(Y\) of cardinality \(\aleph_\sigma\). Its closure \((Y)[X_0]=X\), by Theorem 2, has weight and cardinality \(\aleph_\sigma\) and is \(\aleph_\sigma\)-bicompact as a closed set of the space \(Y\). By Theorem 1, \(T^\sigma X\) is \(\aleph_\sigma\)-bicompact, and since the weight at all points of \(X\) is less than \(\aleph_\sigma\), all points of \(T^\sigma X\) are isolated. Consequently, \(T^\sigma X\) has a disjoint cover of cardinality \(\aleph_\sigma\) by all one-point sets, from which no proper subcover can be selected at all, which contradicts the \(\aleph_\sigma\)-bicompactness of \(T^\sigma X\).

Received
17 VI 1966

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