UDC 519.50+50.01

MATHEMATICS

I. I. PAROVICHENKO

ON TOPOLOGICAL EQUIVALENTS OF THE BRANCHING HYPOTHESIS*

(Presented by Academician P. S. Aleksandrov on 24 IV 1968)

1. The aim of the present paper is to strengthen the results of our preceding paper \((^1)\), consisting in the finding of new topological equivalents of the set-theoretic condition \((\alpha)\) for a strongly inaccessible cardinal number, which we call the branching hypothesis for this number.

Below, as in \((^1)\), \(\aleph_\sigma\) is strongly inaccessible and \(\mathfrak m\) is an infinite cardinal number. We take the remaining terms and notation from \((^1)\) without qualification, and add the following new ones. A system of sets \(\Sigma\) is called an open \(\mathfrak m\)-subbase of a topological space if the collection of intersections of all subsystems in \(\Sigma\) of cardinality \(<\mathfrak m\) forms an open base of this space. A partially ordered set \(M\) is called \(\mathfrak m\)-directed if, for each of its subsets of cardinality \(<\mathfrak m\), there exists an upper bound in \(M\). In particular, \(\aleph_0\)-directed sets are directed sets in the usual sense of the word. If \(X\) is a topological space, then \(\exp(\mathfrak m, X)\) denotes the space on the family of all nonempty closed subsets of \(X\), whose topology is given by the open base consisting of sets of the form \(B(G_0,\mathfrak G)\), where, for an open \(G_0\) and a family \(\mathfrak G\) of cardinality \(<\mathfrak m\) of open sets in \(X\),
\[ B(G_0,\mathfrak G)=\{F\mid F\subseteq G_0,\ \forall G\in\mathfrak G,\ F\cap G\supset \Lambda\}. \]
In particular,
\[ B(G_0,\{G_1,\ldots,G_n\})=B(G_0,G_1,\ldots,G_n) \]
from \((^2)\), p. 168, and for \(\mathfrak m=\aleph_0\) we have the usual Vietoris exponential space (see, for example, \((^2)\) or \((^3)\)).

We formulate the following properties of the number \(\aleph_\sigma\), supplementing the list from \((^1)\):

\(\gamma_T(\gamma^+)\). The Tikhonov \((\mathcal T_\sigma-)\) product of \(\aleph_\sigma\)-bicompact spaces of weight \(\leq \aleph_\sigma\), taken in number \(\leq \aleph_\sigma\), is \(\aleph_\sigma\)-bicompact (analogues of Tikhonov’s theorem).

\(\delta\). The assertion of Theorem 1 from \((^1)\).

\(\delta^+\). If from every cover of a space \(X\) by sets of some open \(\aleph_\sigma\)-subbase of it of cardinality \(\aleph_\sigma\) one can extract a subcover of cardinality \(<\aleph_\sigma\), then \(X\) is \(\aleph_\sigma\)-bicompact (an analogue of Aleksandrov’s lemma).

\(\varepsilon\). If \(X\) is an \(\aleph_\sigma\)-bicompact space of weight \(\aleph_\sigma\), then \(\exp(\aleph_\sigma,\mathcal T_\sigma X)\) is also an \(\aleph_\sigma\)-bicompact space of weight \(\aleph_\sigma\) (an analogue of Vietoris’s theorem, \((^3)\), p. 161).

\(\zeta\). If \(\{X_\lambda,\pi_\mu^\lambda\mid \operatorname{card}\{\lambda\}=\aleph_\sigma\}\) is an inverse \(\aleph_\sigma\)-directed spectrum of nonempty \(\aleph_\sigma\)-bicompact \(\mathcal T_2\)-spaces of weight \(\leq \aleph_\sigma\), then its limit is nonempty (an analogue of Steenrod’s theorem \((^4)\), p. 51).

Theorem. For strongly inaccessible \(\aleph_\sigma\), the conditions \(\beta,\ \beta^+,\ \gamma,\ \gamma_T,\ \gamma^+,\ \delta,\ \delta^+,\ \varepsilon\), and \(\zeta\) are equivalent to the branching hypothesis \(\alpha\).

Since the equivalence of \(\alpha,\ \beta,\ \beta^+,\ \gamma\), and \(\delta\) was proved in \((^1)\), it is necessary to prove only the remaining implications. Obviously, \(\delta^+\Rightarrow\delta\), and the implication \(\alpha\Rightarrow\delta^+\) can be proved without changing anything in the method of proof of \(\alpha\Rightarrow\delta\)

* The results of the paper were reported on 15 II 1968 at a scientific session of the faculty of the Kishinev State University.

(theorem 1 of \((^1)\)). We shall derive the rest according to the scheme:
\(\delta^+ \Rightarrow \varepsilon \Rightarrow \gamma \Rightarrow \alpha \Rightarrow \delta^+ \Rightarrow \nu^\tau \Rightarrow \nu_\tau \Rightarrow \xi \Rightarrow \alpha\). In the proof of \(\delta^+ \Rightarrow \varepsilon\) we used Frink’s idea (see, for example, \((^3)\), p. 161), and in the proof of \(\varepsilon \Rightarrow \gamma\) the idea of Maryanovich \((^5)\) in similar situations.

1. Let \(X\) be a topological space; \(F_0, F_1\) closed in \(X\); put
   \[ \Phi(F_0,F_1)=\{F\mid F\cap F_0=\Lambda \text{ or } \Lambda\subset F\subset F_1\}, \]
   where \(F\) is closed in \(X\). A system of sets shall be called \(\mathfrak m\)-additive (\(\mathfrak m\)-multiplicative) if it contains all unions (intersections) of its subsystems of cardinality less than \(\mathfrak m\).

(a) If \(F_0'\cap F_1''=\Lambda\), then
\[ \Phi(F_0',F_1')\cap \Phi(\Lambda,F_1'')=\Phi(\Lambda,F_1'\cap F_1''). \]
Let \(F\) belong to the left-hand side; then \(F\in\Phi(\Lambda,F_1'')\), whence \(F\subset F_1''\); since \(F_0'\cap F_1''=\Lambda\), \(F\cap F_0'=\Lambda\). But \(F\in\Phi(F_0',F_1')\), whence \(F\subset F_1'\). Thus \(F\subset F_1'\cap F_1''\), and \(F\) belongs to the right-hand side. The reverse inclusion is obvious. It is also obvious that

(b)
\[ \bigcap_\lambda \Phi(\Lambda,F_1^\lambda)=\Phi\left(\Lambda,\bigcap_\lambda F_1^\lambda\right). \]

(c) Let \(\mathfrak m\) be regular and \(X\) an arbitrary \(\mathfrak m\)-bicompact space. Then any \(\mathfrak m\)-centered system of families of closed subsets of \(X\) of the form \(\Phi(F_0,F_1)\) has nonempty intersection.

Let \(\widetilde\Phi\) be our system. Extend \(\widetilde\Phi\) to \(\widetilde\Phi^+\) by adding all sets of the form \(\Phi(F_0,F_1)\) which are intersections of subsystems in \(\widetilde\Phi\) of cardinality \(<\mathfrak m\). Then, by regularity of \(\mathfrak m\), \(\widetilde\Phi^+\) contains all \(\Phi(F_0,F_1)\) which are intersections of subsystems in \(\widetilde\Phi^+\) of cardinality \(<\mathfrak m\) (in particular, \(\Phi(\Lambda,X)\), the intersection of the empty subsystem) and does not contain the empty family \(\Lambda=\Phi(\Lambda,\Lambda)\). We shall prove that even \(\bigcap\widetilde\Phi^+\supset\Lambda\). Let
\[ \widetilde\Phi^+=\{\Phi(F_0^\alpha,F_1^\alpha)\}. \]
Let \(\Psi\) be the family of those \(F_1^\alpha\) for which \(F_0^\alpha=\Lambda\). Then, by (b), \(\Psi\) is \(\mathfrak m\)-multiplicative. Put \(F=\bigcap\Psi\) and prove that \(F\in\bigcap\widetilde\Phi^+\), i.e. \(\forall\alpha\; F\in\Phi(F_0^\alpha,F_1^\alpha)\). Fix \(\alpha\); if \(F_0^\alpha\cap F\supset\Lambda\), then \(F\subset\Phi(F_0^\alpha,F_1^\alpha)\), so let further \(F_0^\alpha\cap F=\Lambda\). We first prove that there exists \(F_1^{\alpha'}\in\Psi\) such that \(F_0^\alpha\cap F_1^{\alpha'}=\Lambda\); for this suppose the contrary: for all \(F_1^{\alpha'}\in\Psi\), \(F_0^\alpha\cap F_1^{\alpha'}\supset\Lambda\). Then the system
\[ \{F_0^\alpha\cap F_1^{\alpha'}\mid F_1^{\alpha'}\in\Psi\} \]
is \(\mathfrak m\)-centered, since \(\Psi\) is \(\mathfrak m\)-multiplicative. But then
\[ \bigcap\{F_0^\alpha\cap F_1^{\alpha'}\mid F_1^{\alpha'}\in\Psi\}=F_0^\alpha\cap F\subset\Lambda \]
by \(\mathfrak m\)-bicompactness of \(X\), which contradicts the initial condition. Thus there exists \(F_1^{\alpha'}\) in \(\Psi\) such that
\[ F_0^\alpha\cap F_1^{\alpha'}=\Lambda. \]
Then by (a)
\[ \Phi(\Lambda,F_1^\alpha\cap F_1^{\alpha'})=\Phi(\Lambda,F_1^{\alpha'})\cap \Phi(F_0^\alpha,F_1^\alpha), \]
and the sets on the right belong to \(\widetilde\Phi^+\), whence \(\Phi(\Lambda,F_1^\alpha\cap F_1^{\alpha'})\in\widetilde\Phi^+\), \(F_1^\alpha\cap F_1^{\alpha'}\in\Psi\), \(F=\bigcap\Psi\subset F_1^\alpha\cap F_1^{\alpha'}\subset F_1^\alpha\). Since \(\Psi\) is \(\mathfrak m\)-multiplicative and does not contain the empty set, \(\Psi\) is \(\mathfrak m\)-centered in the \(\mathfrak m\)-bicompact space \(X\), so that \(F_1^\alpha\supset F=\bigcap\Psi\supset\Lambda\) and \(F\in\Phi(F_0^\alpha,F_1^\alpha)\).

(d) If \(\mathfrak B\) is an \(\mathfrak m\)-additive open base of the \(\mathfrak m\)-bicompact space \(X\), then
\[ \{B(H_0,\mathfrak H)\mid H_0\in\mathfrak B,\ \mathfrak H\subseteq\mathfrak B,\ \operatorname{card}\mathfrak H<\mathfrak m\} \]
is an open base in \(\exp(\mathfrak m,X)\).

Let \(B(G_0,\mathfrak G)\) be a neighborhood of \(F\) in \(\exp(\mathfrak m,X)\). Take a covering of \(F\) by sets from \(\mathfrak B\) lying in \(G_0\), and select from it a subcover of cardinality \(<\mathfrak m\); the union of the latter, by \(\mathfrak m\)-additivity of \(\mathfrak B\), is some \(H_0\in\mathfrak B\), and \(F\subseteq H_0\subseteq G_0\). Further, for each \(G\in\mathfrak G\) take \(x(G)\in G\cap F\) and in \(\mathfrak B\) choose \(H(G)\), \(x(G)\in H(G)\subseteq G\). Denote the collection of all such \(H(G)\) by \(\mathfrak H\). Then
\[ F\in B(H_0,\mathfrak H)\subseteq B(G_0,\mathfrak G), \]
as was required here.

(e) If \(X\) has weight \(\aleph_\sigma\), then \(\exp(\aleph_\sigma,X)\) has weight \(\ge \aleph_\sigma\).

Since \(X\) has weight \(\aleph_\sigma\), there are at least \(\aleph_\sigma\) closed sets in \(X\). But from \(F_1\setminus F_2\supset\Lambda\) it follows that
\[ F_1\in B(X,CF_2)\setminus B(X,CF_1), \]
so that there are at least \(\aleph_\sigma\) open sets in \(\exp(\aleph_\sigma,X)\). And a space of weight \(\aleph_\nu<\aleph_\sigma\) has \(\le 2^{\aleph_\nu}<\aleph_\sigma\) open sets.

From (d) and (e) there follows, independently of the branching hypothesis, the assertion part of \((\varepsilon)\):

(f) If \(X\) is an \(\aleph_\sigma\)-bicompact space of weight \(\aleph_\sigma\), then \(\exp(\aleph_\sigma,X)\) has weight \(\aleph_\sigma\).

Since \(\sum_{\alpha<\sigma}\aleph_\sigma^{\aleph_\alpha}=\aleph_\sigma\), for the proof it is enough to enlarge an open base of \(X\) of cardinality \(\aleph_\sigma\) to an equipotent \(\aleph_\sigma\)-additive one by adding the unions of all its subsystems of cardinality \(<\aleph_\sigma\).

(g) \(\displaystyle \bigcap_\lambda B(G_0^\lambda,\mathfrak G^\lambda)=B\left(\bigcap_\lambda G_0^\lambda,\bigcup_\lambda \mathfrak G^\lambda\right)\); in particular, \(\displaystyle \bigcap_\lambda B(G_0,G_1^\lambda)=B(G_0,\{G_1^\lambda\})\).

1. \((\delta^+)\Rightarrow(\varepsilon)\). Let \(X\) be an \(\aleph_\sigma\)-bicompact space of weight \(\aleph_\sigma\); by \((\delta)\), \(\mathcal T^\sigma X\) is also an \(\aleph_\sigma\)-bicompact space of weight \(\aleph_\sigma\), and let \(\mathfrak B\) be an \(\aleph_\sigma\)-additive open base of \(\mathcal T^\sigma X\) of cardinality \(\aleph_\sigma\) (cf. the proof of (f)). Then, by (d), \(\{B(H_0,\mathfrak H)\mid H_0\in\mathfrak H,\ \mathfrak H\subseteq\mathfrak B,\ \operatorname{card}\mathfrak H<\aleph_\sigma\}\) is an open base of \(\exp(\aleph_\sigma,\mathcal T^\sigma X)\) of cardinality \(\aleph_\sigma\). Since \(\mathcal T^\sigma X\) is a \(\mathcal T^\sigma\)-space, by (g) the family \(\{B(H_0,H_1)\mid H_0,H_1\in\mathfrak B\}\) forms an open \(\aleph_\sigma\)-subbase of \(\exp(\aleph_\sigma,\mathcal T^\sigma X)\) of cardinality \(\aleph_\sigma\), and by \((\delta^+)\), passing to complements \(CB(H_0,H_1)=\Phi(CH_0,CH_1)\), and using (c), we obtain the required assertion.

\((\varepsilon)\Rightarrow(\gamma)\). Replacing, for the disjoint sum \(\cup\), the sign \(\Pi\), introduce the topological sum \(\displaystyle \Pi_\lambda D_\lambda=S\), where the \(D_\lambda\) are two-point discrete spaces, \(\operatorname{card}\{\lambda\}=\aleph_\sigma\), and consider its one-point extension \(S^+=S\Pi\infty\), where neighborhoods of \(\infty\) are taken to be \(O(\infty)=\infty\Pi M\), \(M\subseteq S\), \(\operatorname{card}CM<\aleph_\sigma\). It is obvious that a set is closed in \(S^+\) if and only if it either contains \(\infty\), or has cardinality \(<\aleph_\sigma\). It is also easy to see that \(S^+\) is an \(\aleph_\sigma\)-bicompact \(\mathcal T^\sigma\)-space of weight \(\aleph_\sigma\), and, by \((\varepsilon)\), \(\exp(\aleph_\sigma,S^+)\) has the same properties. Consider in \(\exp(\aleph_\sigma,S^+)\) points of the form \(\displaystyle \Pi_\lambda x_\lambda \Pi \infty\), where \(x_\lambda=x(\lambda)\) is a choice function for the whole family \(\{D_\lambda\}\); denote all such points by \(\Xi\) and prove that \(\Xi\) is closed in \(\exp(\aleph_\sigma,S^+)\). Let \(F\in C\Xi\); then for \(F\) there are two possibilities: (1) \(\exists\lambda_0,\ F\cap D_{\lambda_0}=\Lambda\); then \(B(CD_{\lambda_0},CD_{\lambda_0})\) is a neighborhood of \(F\) not meeting \(\Xi\); (2) \(\exists\lambda_0,\ F\supset D_{\lambda_0}\), \(D_{\lambda_0}=\{x_{\lambda_0},y_{\lambda_0}\}\); then \(B(S^+,\{x_{\lambda_0}\},\{y_{\lambda_0}\})\) is again a neighborhood of \(F\) not meeting \(\Xi\). Since \(S^+\) is \(\aleph_\sigma\)-bicompact, \(\Xi\) is also \(\aleph_\sigma\)-bicompact. But the correspondence \(\displaystyle \Pi_\lambda x_\lambda\Pi\infty\to\{x_\lambda\}\) is a natural homeomorphism of \(\Xi\) onto the \(\mathcal T^\sigma\)-product of the family \(\{D_\lambda\}\), whence everything follows.

\((\delta^+)\Rightarrow(\gamma^+)\Rightarrow(\gamma_T)\). Let \(P=\Pi^\sigma X_\lambda\) be our \(\mathcal T^\sigma\)-product; obviously, it is enough to prove the \(\aleph_\sigma\)-bicompactness of the product \(P^\sigma=\Pi^\sigma\mathcal T^\sigma X_\lambda\) with the stronger topology. By \((\delta)\), the \(\mathcal T^\sigma X_\lambda\) are also \(\aleph_\sigma\)-bicompact and have weight \(\leq\aleph_\sigma\). Let \(\pi_\lambda\) be the projection of \(P^\sigma\) onto \(\mathcal T^\sigma X_\lambda\), and let \(\mathfrak B_\lambda\) be open bases in \(\mathcal T^\sigma X_\lambda\) of cardinality \(\leq\aleph_\sigma\). Since the \(\mathcal T^\sigma X_\lambda\) are \(\mathcal T^\sigma\)-spaces, \(\{\pi_\lambda^{-1}(B_\lambda)\mid B_\lambda\in\mathfrak B_\lambda,\ \lambda\in\{\lambda\}\}\) is an open \(\aleph_\sigma\)-subbase of \(P^\sigma\) (of cardinality \(\leq\aleph_\sigma\)). The rest of the argument is carried out analogously to \((^6)\), p. 143, with the obvious modification. Since the Tikhonov product is a compactification of the corresponding \(\mathcal T^\sigma\)-product, \((\gamma^+)\Rightarrow(\gamma_T)\).

\((\gamma_T)\Rightarrow(\zeta)\). Let \(\{X_\lambda,\pi_\mu^\lambda\}\) be our spectrum. The \(\aleph_\sigma\)-centeredness of the system \(\{S_\mu^\lambda\mid \lambda>\mu\}\), \(S_\mu^\lambda=\{x\mid x\in\Pi X_\lambda,\ \pi_\mu^\lambda(x_\lambda)=x_\mu\}\), is proved analogously to the corresponding centeredness in \((^4)\), p. 51. Since \(\Pi X_\lambda\) is \(\aleph_\sigma\)-bicompact, \(\bigcap_{\lambda>\mu}S_\mu^\lambda\supset\Lambda\), as was required.

\((\zeta)\Rightarrow(\alpha)\). Let \(S\) be a branching system of order \(\omega_\sigma\). Let \(x\in S\); denote by \(p(x)\) the order of \(x\) in \(S\) and put

\[ S_\lambda=\{x\mid x\in S,\ p(x)=\lambda\},\qquad \tau(x)=\sup\{p(y)\mid y\geq x\text{ in }S\}. \]

It is easy to verify the following properties of \(\tau(x)\): (1) if \(y<x\) in \(S\), then \(\tau(y)\geq\tau(x)\), and (2) if \(\tau(y)\geq\lambda,\ y\in S_\mu,\ \mu<\lambda\), then \(\tau(y)=\sup\{\tau(v)\mid v\in S_\lambda,\ v\geq y\text{ in }S\}\). Put \(S_\lambda^0=\{x\mid x\in S_\lambda,\tau(x)=\omega_\sigma\}\) and prove that all these sets are nonempty. Indeed, otherwise there exists \(\lambda_0<\)

\(< \omega_\sigma\) such that for all \(x \in S_{\lambda_0}\), \(\tau(x) < \omega_\sigma\). Since \(\operatorname{card} S_{\lambda_0} < \aleph_\sigma\), in view of the regularity of \(\omega_\sigma\) we have \(\sup\{\tau(x)\mid x \in S_{\lambda_0}\} < \omega_\sigma\), and the order of the entire system is \(< \omega_\sigma\). We now define, for \(\lambda > \mu\), \(x \in S_\lambda^0\), \(\pi_\mu^\lambda x = y\), where \(y \in S_\mu\) and \(y \le x\) in \(S\). Such a \(y\) exists and is unique by the definition of a branching system. By property (1), for \(\tau(x)\) the mapping \(\pi_\mu^\lambda\) maps \(S_\lambda^0\) into \(S_\mu^0\), and we shall prove that this mapping is onto.

Suppose the contrary: there exists \(z \in S_\mu^0\), \(z \notin \pi_\mu^\lambda(S_\lambda^0)\). Since \(\tau(z) = \omega_\sigma > \lambda\), by property (2)
\[ \tau(z)=\sup\{\tau(v)\mid v \in S_\lambda \setminus S_\lambda^0,\ v \ge z \text{ in } S\}. \]
Since all \(\tau(v) < \omega_\sigma\) and \(\operatorname{card} S_\lambda < \omega_\sigma\), we have \(\tau(z) < \omega_\sigma\), which is impossible. Considering now all \(S_\lambda^0\) with the discrete topologies, we obtain an inverse spectrum \(\{S_\lambda^0,\pi_\mu^\lambda\}\) satisfying condition \((\zeta)\). An element of its nonempty limit gives the sequence in \(S\) required by condition \((\alpha)\).

Remark. Since assertion (c) of item 2 is proved without using the axiom of choice, Vietoris’ theorem on the bicompactness of the ordinary exponential space over a bicompact space is obtained by using only Alexander’s lemma. Since, by Marjanović’s method, the Tikhonov product \(\prod_\lambda X_\lambda\) of bicompact \(\mathcal{T}_2\)-spaces is embedded as a closed set in the exponential space over the one-point bicompactification of P. S. Aleksandrov of the topological sum \(\prod_\lambda X_\lambda\), it follows from Alexander’s lemma that Tikhonov’s theorem already holds for \(\mathcal{T}_2\)-multipliers. This method does not work in the case of \(\mathcal{T}_2\)-multipliers, since then the corresponding set in the exponential space ceases to be closed, which was to be expected, since according to Kelley’s theorem (7) the case of \(\mathcal{T}_1\)-multipliers is already exactly equivalent to the axiom of choice (cf. the comments in this connection in (8), p. 274).

Kishinev State
University

Received
19 IV 1968

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