B. Efimov

On Dyadic Bicompacta

(Presented by Academician P. S. Aleksandrov on 3 XI 1962)

§ 1. Auxiliary propositions.

Let

\[ D^\tau=\prod_{\alpha\in\theta} D_\alpha^{0,1},\qquad \operatorname{card}\theta=\tau, \]

where \(D_\alpha^{0,1}\) is a space consisting of two isolated points, and \(\prod\) denotes topological product. Let \(w\subset\theta\). We shall call the set \(H_w^{i(w)}\) of those points \(\{x_\alpha\}\in D^\tau\) for which, for all \(\alpha\in w\), we have \(x_\alpha=i_\alpha\in i(w)\), while the remaining coordinates are arbitrary, a layer of the space \(D^\tau\) with base \(w\). Let \(F\subset D^\tau\) be closed, and let \(\chi F=\tau_0\leqslant\tau\) be the least cardinal number for which there exists a system \(\{H_\nu\}\) of cardinality \(\tau_0\) of open-and-closed sets with intersection \(\bigcap_\nu H_\nu=F\). This number \(\chi F\) will be called the neighborhood character of the set \(F\) (in the space \(D^\tau\)).

It can be proved that each open-and-closed set \(H_\nu\) is the sum of a finite number of elementary neighborhoods with one and the same base; therefore

\[ F=\bigcap_\nu\left(\bigcup_{s=1}^{k(\nu)} H_{\lambda_\nu}^{s}\right) =\bigcup_\xi\bigcap_{\nu,s} H_{\lambda_\nu}^{s},\qquad 1\leqslant s\leqslant k(\nu). \tag{1} \]

Here by \(\xi\) are meant all possible nonempty intersections formed as follows: from each open-and-closed \(H_\nu\) an arbitrary neighborhood \(H_{\lambda_\nu}^{s}\), \(1\leqslant s\leqslant k(\nu)\), is taken, and the intersection of all selected neighborhoods is considered. Since an elementary neighborhood is the intersection of a finite number of one-index neighborhoods, we have

\[ \bigcap_{\nu,s} H_{\lambda_\nu}^{s} = \bigcap_\nu \left( H_{\alpha_{1\nu}}^{i_{1\nu}}\cap\cdots\cap H_{\alpha_{t(s),\nu}}^{i_{t(s),\nu}} \right) = \bigcap_\mu H_{\alpha_\mu}^{i_\mu} = H_w^{i(\xi)}, \tag{2} \]

where \(w=\bigcup_\mu \alpha_\mu\), and \(i(\xi)\) is the corresponding collection of zeros and ones, depending on \(\xi\). Thus \(H_w^{i(\xi)}\) will be a layer of \(D^\tau\) with base \(w\), where \(\operatorname{card} w\leqslant\tau_0\). Combining (1) and (2), we obtain that \(F=\bigcup_\xi H_w^{i(\xi)}\), with \(\operatorname{card} w\leqslant\chi F\).

Every decomposition of a closed set \(F\) into \(\bigcup_\xi H_{w(\xi)}^{i(\xi)}\) will be called a stratification of \(F\).

We have now proved the following proposition:

A. If \(\chi F=\tau_0\), then there exists a stratification of \(F\), the cardinality of the base of each layer of which does not exceed the neighborhood character of \(F\).

It can be shown that if the base \(w\) is one and the same for each layer, then the stratification is a continuous partition of the set \(F\). The space of this partition will be called the skeleton of \(F\) (\(\operatorname{sk} F\)). In this case we have the following two properties of \(\operatorname{sk} F\).

B. The weight of the skeleton does not exceed the neighborhood character of \(F\).

C. If at least one skeleton of the bicompactum \(F\) is dyadic, then the bicompactum \(F\) itself is dyadic (and then each of its skeletons is dyadic).

We now prove the following proposition.

D. If \(F=\left[\bigcup_\xi H_{w(\xi)}^{i(\xi)}\right]\), where \(\operatorname{card} w_\xi\leqslant\tau_0\), then \(\chi F\leqslant\tau_0\).

Proof. We may suppose that \(D^\tau \setminus F \ne \Lambda\). For any point \(x \in D^\tau \setminus F\) consider
\[ H_x=H^{i_1\ldots i_s}_{a_1\ldots a_s}\subset D^\tau \setminus F. \]
Since
\[ H^{i_1\ldots i_s}_{a_1\ldots a_s}\cap H_w^{i(\xi)}(\xi)=\Lambda \]
for any \(\xi\), this means that for each \(\xi\) we have
\[ e=a_1,\ldots,a_s\cap w(\xi)\ne\Lambda \]
and there is at least one \(a_j=a_\mu^\xi\in e\) such that
\[ i_{a_j}\ne i_{a_\mu^\xi}. \]
Call this condition (L). A neighborhood
\[ H^{i_1\ldots i_s}_{a_1\ldots a_s} \]
satisfying condition (L) will be called regular if there is no proper part \(a_1,\ldots,a_k\subset a_1,\ldots,a_s\) such that the neighborhood
\[ H^{i_1\ldots i_k}_{a_1\ldots a_k} \]
satisfies condition (L). It can be shown that every point \(x\in D^\tau\setminus F\) has a regular neighborhood. If we show that the cardinality of the set of distinct regular neighborhoods does not exceed \(\tau_0\), then it will thereby be proved that
\[ \chi F\le \tau_0. \]
Denote this family by
\[ H=\{H^{i_\mu}_{\lambda_\mu}\}. \]

Suppose the contrary. Let \(\tau_0=\aleph_\sigma\); then \(\operatorname{card} H\ge \aleph_{\sigma+1}\) is uncountable. Therefore there exists a subfamily of the family \(H\) (which we shall still denote by
\[ H=\{H^{i_\mu}_{\lambda_\mu}\}), \]
all neighborhoods of which have one and the same rank \(s\), and moreover
\[ \operatorname{card} H\ge \aleph_{\sigma+1}. \]
Consider some layer
\[ H^{i(\xi_0)}_{w(\xi_0)}\subset F. \]
From condition (L) it follows that for each neighborhood \(H^{i_\mu}_{\lambda_\mu}\in H\),
\[ \lambda_\mu\cap w(\xi_0)\ne\Lambda. \]
To each
\[ a_\nu^\xi\in w(\xi_0) \]
assign the cardinality of those \(H^{i_\mu}_{\lambda_\mu}\in H\) whose bases contain the index \(a_\nu^\xi\). Since
\[ \operatorname{card} w(\xi)\le \aleph_\sigma, \]
and
\[ \operatorname{card} H\ge \aleph_{\sigma+1}, \]
there exists a subfamily \(H_1\) of the family \(H\), with
\[ \operatorname{card} H_1\ge \aleph_{\sigma+1}, \]
all bases of whose neighborhoods contain one and the same index \(a_1\), and the \(a_1\)-st coordinate is the same in all neighborhoods. Next we continue by induction. Suppose that we have found a subfamily \(H^{n-1}\) of the family \(H\), with
\[ \operatorname{card} H^{\,n-1}\ge \aleph_{\sigma+1}, \]
all bases of whose neighborhoods contain the same indices \(a_1,\ldots,a_{n-1}\), and the values
\[ i_{a_1},\ldots,i_{a_{n-1}} \]
coincide in all neighborhoods of the family \(H^{n-1}\). Consider an arbitrary representative of this family
\[ H_\lambda= H^{i_{a_1}\ldots i_{a_{n-1}}\,i_{a_n}\ldots i_{a_s}}_{a_1\ldots a_{n-1}\,a_n\ldots a_s}. \]
The neighborhood
\[ H^{i_{a_1}\ldots i_{a_{n-1}}\,i_{a_{n+1}}\ldots i_{a_s}}_{a_1\ldots a_{n-1}\,a_{n+1}\ldots a_s}, \]
which is obtained from \(H_\lambda\) by omitting the index \(a_n\), is no longer regular; consequently, there exists a layer
\[ H^{i(\xi_0)}_{w(\xi_0)}\subset F, \]
for which either 1)
\[ a_1,\ldots,a_{n-1},a_{n+1},\ldots,a_s\cap w(\xi_0)=\Lambda \]
or 2)
\[ a_1,\ldots,a_{n-1},a_{n+1},\ldots,a_s\cap w(\xi_0)\ne\Lambda \]
and for all \(a_j=a_\nu^{\xi_0}\) we have
\[ i_{a_j}=i_{a_\nu^{\xi_0}},\qquad a_\nu^{\xi_0}\in w(\xi_0). \]
In case 1), every \(\lambda_\mu\) intersects \(w(\xi_0)\) in indices not belonging to \(a_1,\ldots,a_{n-1}\). Since
\[ \operatorname{card} w(\xi_0)\le \aleph_\sigma \]
and
\[ \operatorname{card} H^{\,n-1}\ge \aleph_{\sigma+1}, \]
there exists a subfamily \(H^n\) of the family \(H^{n-1}\), with
\[ \operatorname{card} H^n\ge \aleph_{\sigma+1}, \]
all bases of whose neighborhoods contain one and the same index
\[ a_\nu^{\xi_0}=a_n \]
and for which \(i_{a_n}\) coincides. In case 2), we again find that every \(\lambda_\mu\) intersects \(w(\xi_0)\) in indices not belonging to \(a_1,\ldots,a_{n-1}\), since every neighborhood
\[ H^{i_\mu}_{\lambda_\mu}\in H^{n-1} \]
satisfies condition (L). Thus, in both cases we obtain that in the family \(H^n\) all bases of neighborhoods contain the same indices
\[ a_1,\ldots,a_n, \]
and all values
\[ i_{a_1},\ldots,i_{a_n} \]
coincide in all neighborhoods. Consequently, after no more than \(s\) steps we obtain that there exists a subfamily \(H^s\) of the family \(H\), with
\[ \operatorname{card} H^s\ge \aleph_{\sigma+1}, \]
all neighborhoods of which coincide, which contradicts the fact that all
\[ H^{i_\mu}_{\lambda_\mu} \]
are distinct. D is proved.

The following propositions easily follow from D.

Theorem 1. A canonical closed set in the space \(D^\tau\) is a set of type \(G_\delta\) (consequently, every canonical open set is a set of type \(F_\sigma\)).

Next we have the proposition:

If the character of a closed set \(F\) is uncountable, then in any partition of \(F\) there exists a layer \(H^{i}_{w}(\xi_0)\) such that, for every countable subset \(v \subset w(\xi_0)\), the layer \(H^{i}_{v}\), containing \(H^{i}_{w(\xi_0)}(\xi_0)\), intersects \(D^\tau \setminus F\).

Now the following known theorem is easily proved:

Theorem 2. The weight of a dyadic bicompactum \(R\) is equal to the least upper bound of the values of the neighborhood characters of the points \(x \in R\).

Indeed, let
\[ \tau_0=\sup_{x\in R}\chi x \]
and \(R=f(D^\tau)\); then, to prove Theorem 2, it is enough to show that \(\chi_{R\times R}\Delta \leq \tau_0\). It is easy to note that \(\chi y\leq \tau_0\) for all \(y\in\Delta\) and that \(R\times R=g(D^\tau)\). Then
\[ \chi_{D^\tau}(g^{-1}\Delta)=\chi_{D^\tau}\!\left(\bigcup_{y\in\Delta}F_y\right), \]
where \(F_y=g^{-1}y\). Since \(\chi F_y\leq \tau_0\), it follows, applying A, that
\[ F_y=\bigcup_{\xi} H^{i}_{w(y)}(\xi), \]
where \(\operatorname{card} w(y)\leq \tau_0\); further,
\[ g^{-1}\Delta=\bigcup_y F_y=\bigcup_y\bigcup_{\xi} H^{i}_{w(y)}(\xi); \]
applying D, we obtain
\[ \chi_{D^\tau}(g^{-1}\Delta)\leq \tau_0, \]
as was required.

§ 2. Main results.

Theorem 3 (the first main theorem). In a dyadic bicompactum every canonical closed set and every closed \(G_\delta\) are dyadic bicompacta.

Proof. Let \(R=f(D^\tau)\); \(F=[U]\subset R\), \(U\) open. Then
\[ F=f(\widetilde F), \]
where
\[ \widetilde F=[f^{-1}U] \]
is canonical closed, which is \(G_\delta\), by Theorem 1. If, on the other hand, \(F\) is a closed \(G_\delta\) in \(R\), then its full preimage will be a closed \(G_\delta\) in \(D^\tau\). Thus, in both cases, the proof reduces to establishing the fact that every closed \(G_\delta\) set \(\widetilde F\) in \(D^\tau\) is dyadic. We show this. Since \(\widetilde F\) is a closed \(G_\delta\) in \(D^\tau\), we have \(\chi\widetilde F\leq \aleph_0\); applying B, we obtain that the weight \(\operatorname{sk} F\leq \aleph_0\); hence, by the theorem of P. S. Aleksandrov \((^1)\), the bicompactum \(\operatorname{sk}\widetilde F\) is dyadic; using proposition C, we obtain that \(\widetilde F\) itself is dyadic. The theorem is proved.

Theorem 4 (the second main theorem). A hereditarily dyadic bicompactum is metrizable.

Proof. Denote by \(E_\tau\) the space of cardinality \(\tau\geq\aleph_0\) consisting of isolated points only, and by \(bE_\tau\) some bicompact extension of \(E_\tau\). In particular, denote by \(b_0E_\tau\) the bicompactum obtained by adjoining to \(E_\tau\) a single point \(\xi\), called the vertex of \(b_0E_\tau\). If \(\tau\geq\aleph_1\), then the bicompactum \(bE_\tau\) is not dyadic \((^2)\). We first prove the following proposition:

E. A continuous dyadic image of \(bE_\tau\), under the condition that the remainder
\[ N=bE_\tau\setminus E_\tau \]
is mapped to a single point, is the bicompactum \(b_0E_{\aleph_0}\).

Let \(Y=f(bE_\tau)\) and \(z=f(N)\in Y\). We show that \(Y=b_0E_{\tau'}\), \(\tau'\leq\tau\), and \(z\) is the vertex of \(Y\). Indeed, let \(Oz\) be an arbitrary neighborhood of the point \(z\); choose \(ON\) so that
\[ f(ON)\subseteq Oz; \]
since outside \(ON\) there lies a finite number of isolated points of \(bE_\tau\), outside \(Oz\) there can lie only the images of these points. Hence it follows at once that \(z\) is the unique non-isolated point of the bicompactum \(Y\). Thus
\[ Y=b_0E_{\tau'}, \]
with \(\tau'\leq\tau\). If, moreover, \(Y\) is a dyadic bicompactum, then
\[ \tau'=\aleph_0. \]

As a consequence of proposition E we obtain that the full preimage of the point \(z\) contains all points of \(E_\tau\), with the exception of at most countably many. Now let
\[ R=f(D^\tau) \]
be a hereditarily dyadic bicompactum. To prove that \(R\) is metrizable, it is enough, by Theorem 2, to show that \(R\) satisfies the first axiom of countability, i.e. that for each

of the point \(x\in R\), \(F=f^{-1}x\) has countable neighborhood character in \(D^\tau\). Suppose that for some point \(x_0\in R\), \(\chi F_0\geq \aleph_1\), where \(F_0=f^{-1}x_0\). We shall then show that in \(D^\tau\) there lies a bicompactum \(bE_{\aleph_1}\) such that
\[ N=bE_{\aleph_1}\setminus E_{\aleph_1}\subset F_0, \]
and moreover \(E_{\aleph_1}\cap F_0=\Lambda\). This means, as was shown above, that \(f(bE_{\aleph_1})=b_0E_{\aleph_1}\), which lies in \(R\), contradicting the hereditary dyadicity of \(R\). We shall construct \(bE_{\aleph_1}\) by transfinite induction. Consider some partition \(F_0\). Using the proposition, consider a layer
\[ U=H_{\mathfrak w(\xi_0)}^{\,i(\xi_0)} \]
such that, for any countable subset \(v\subset \mathfrak w(\xi_0)\), the layer \(H_v^{\,i(v)}\) containing \(U\) intersects \(D^\tau\setminus F_0\). Let \(\alpha_1\in \mathfrak w(\xi_0)\). Take \(H_{\alpha_1}^{\,i_{\alpha_1}}\supset U\). There will be found a point \(y_1\in H_{\alpha_1}^{\,i_{\alpha_1}}\) not belonging to \(F_0\). Consider
\[ H_{\lambda_1}= H_{\beta_1\ldots\beta_{s(1)}}^{\,i_1\ldots i_{s(1)}}(y_1)\subset D^\tau\setminus F_0. \]
From \(H_{\lambda_1}\) take a point \(x_1\) such that on the indices \(\mathfrak w(\xi_0)\setminus\lambda_1\) it assumes the very same values as the layer \(U\), and on all the remaining, non-fixed ones, zeros. Denote by \(A_1\) the set of those indices \(\beta\in\lambda_1\) which enter into \(\mathfrak w(\xi_0)\) and carry opposite values. Such indices necessarily exist. Suppose that, for an arbitrary countable ordinal \(v\), the points \(x_1,\ldots,x_\mu,\ldots\), \(\mu<v\), and the sets \(A_\mu\) have been constructed. Denote
\[ A_v^*=\bigcup_{\mu<v} A_\mu . \]
It is easy to see that \(\operatorname{card} A_v^*\leq \aleph_0\), and that \(A_v^*\subset \mathfrak w(\xi_0)\); therefore, for the layer
\[ H_{A_v^*}^{\,i(A_v^*)}\supset U \]
there will be found a point \(y_v\in D^\tau\setminus F_0\). Consider
\[ H_{\lambda_v}=H_{\beta_1^v\ldots \beta_{s(v)}^v}^{\,i_1\ldots i_{s(v)}}(y_v)\subset D^\tau\setminus F_0. \]
From \(H_{\lambda_v}\) take a point \(x_v\) such that on the indices \(\mathfrak w(\xi_0)\setminus\lambda_v\) it assumes the very same values as the layer \(U\), and on all the remaining non-fixed ones, zeros. Just as at the first step, from \(\lambda_v\) select the set \(A_v\). The induction is complete. We shall prove that \(\{x_v\}=bE_{\aleph_1}\) is the required bicompactum. Note that \(A_v\cap A_\mu=\Lambda\). Indeed, let \(\mu<v\); then \(A_\mu\subset A_v^*=\bigcup_{\gamma<v}A_\gamma\), and therefore the point \(x_v\) on the indices \(A_\mu\) carries the very same values as the layer \(U\), while on the indices \(A_v\) it carries opposite values; consequently, \(A_v\) and \(A_\mu\) do not intersect. It follows from this that the neighborhood
\[ H_{A_v}^{\,i(A_v)} \]
does not contain a single point \(x_\mu\), since \(x_\mu\), on the indices \(\mathfrak w(\xi_0)\setminus A_\mu\) containing \(A_v\), carries the same values as the layer \(U\), and hence values opposite to the indices \(i(A_v)\). Thus \(\{x_v\}=E_{\aleph_1}\). We now show that no point of \(D^\tau\setminus U\) is a limit point for \(\{x_v\}\). If \(y\notin U\), this means that there is an index \(\alpha\in\mathfrak w(\xi_0)\) such that \(y_\alpha\ne i_\alpha\), where \(i_\alpha\) is the corresponding value of the layer \(U\). In that case the neighborhood
\[ H_\alpha^{\,y_\alpha}(y) \]
can contain no more than one point of the set \(\{x_v\}\), since there exists at most one \(A_v\ni\alpha\). Thus the remainder \([\{x_v\}]\) lies entirely in the layer \(U\), and consequently in \(F_0\). The theorem is proved.

A strengthening of Theorem 3 is the following theorem:

Theorem 5. If \(\tau\geq\aleph_0\), then every nonempty canonical closed set of the space \(D^\tau\) is homeomorphic to the whole \(D^\tau\). If \(\tau\geq\aleph_1\), then every closed \(G_\delta\) in \(D^\tau\) is homeomorphic to the whole \(D^\tau\).

The author expresses his gratitude to P. S. Aleksandrov and V. I. Ponomarev for valuable suggestions.

Moscow State University
named after M. V. Lomonosov

Received
31 X 1962

CITED LITERATURE

1. P. S. Aleksandrov, Introduction to the General Theory of Sets and Functions, 1958.
2. E. Marczewski, Fund. Math., 34, 127 (1947).