UDC 513.83

MATHEMATICS

V. V. FILIPPOV

A BICOMPACTUM WITH NONCOINCIDING INDUCTIVE DIMENSIONS

(Presented by Academician P. S. Aleksandrov, 19 XI 1968)

P. S. Aleksandrov posed the question of the coincidence of the dimensional invariants $\operatorname{ind}$ and $\operatorname{Ind}$ for bicompacta. The present note is devoted to the construction of a bicompactum $X$ with $\operatorname{ind} X = 2$, $\operatorname{Ind} X = 3$.

Let $Y_1$ be the set of sequences $\{x_\alpha\}_{\alpha<\omega_1}$, enumerated by transfinite numbers (strictly) less than $\omega_1$, of points of the segment $I=[0,1]$, such that if at some place in the sequence there stands $0$ or $1$, then in all subsequent places the same number stands. The set $Y_1$ carries the natural lexicographic order, i.e., $\{y_\alpha\}<\{x_\alpha\}$ if $y_\beta<x_\beta$, where $\beta$ is the least of the transfinite numbers at which the sequences do not coincide. It is not difficult to verify that the space $Y_1$ in the order topology is a connected bicompactum. We may represent the space $Y_1$ as the union of two everywhere dense subsets $Y_1^0$ and $Y_1^1$, where $Y_1^1$ consists of points of countable character (those sequences in which, beginning with some place, there stands $0$ or $1$), and $Y_1^1$ consists of points of uncountable ($\aleph_1$) character, which are not limit points for any countable set (those sequences in which there are no $0$ and $1$). The family of all intervals of the form $(r_1,r_2)$ and half-intervals of the form $[\min Y_1,r_2)$ and $(r_1,\max Y_1]$, where $r_1,r_2\in Y_1^0$, forms a base of the space $Y_1$. Let $\Xi$ be the family of those ordered pairs of elements of this base in which the second member lies with its closure in the first and both are simultaneously either intervals or half-intervals. The cardinality of the family $\Xi$ is equal to the cardinality of the set $Y_1^0$, which is equal to $c^{\aleph_0}\cdot\aleph_1=c$. The intersection of $Y_1^1$ with any open set has cardinality $c^{\aleph_1}\ge c$. By a theorem of Sierpiński (1), the subspace $Y_1^1$ can be represented as the union of $\ge c$ subsets everywhere dense in $Y_1^1$ and, consequently, everywhere dense in $Y_1$. Let $R$, $P_\alpha^\xi$ $(\xi\in\Xi,\alpha<\omega_1)$ be pairwise disjoint everywhere dense subsets of the set $Y_1^1$, the family $\{P_\alpha^\xi\}$ of such subsets being indexed by transfinite numbers less than $\omega_1$ and by elements of the family $\Xi$. Similarly, from $Y_1^0$ we can single out a family $\{Q_\alpha^\xi\}_{\xi\in\Xi,\alpha<\omega_1}$ of pairwise disjoint sets everywhere dense in $Y_1$.

Fix $\alpha<\omega_1$. To an element $\xi$ of the family $\Xi$ we put in correspondence the set $b_\xi^\alpha$, which is an interval $(y_1,y_2)$ if $\xi$ consists of intervals, and a half-interval $[\min Y_1,y_2)$ or $(y_1,\max Y_1]$ if $\xi$ consists of half-intervals, the set $b_\xi^\alpha$ being chosen in such a way that it lies in the first member of the pair $\xi$ and contains the second, and $y_1,y_2\in P_\alpha^\xi$. The resulting family $B_\alpha=\{b\}_{\xi\in\Xi}$ is, as is easy to see, a base of the space $Y_1$.

Thus, we have obtained $\aleph_1$ bases $B_\alpha$, $\alpha<\omega_1$, of the space $Y_1$. For us it will be essential that no two elements of the family $\bigcup_{\alpha<\omega_1} B_\alpha$ have common open ends (this follows from the fact that the sets $P_\alpha^\xi$ are pairwise disjoint).

Take some point $\vartheta\in Y_1^1\setminus R$. Fix a sequence

the family \(\{H_\alpha\}_{\alpha<\omega_1}\), indexed by transfinite numbers less than \(\omega_1\), containing the point \(\vartheta\), of intervals contracting to \(\vartheta\) such that \([H_\alpha]\subset H_\beta\) if \(\alpha>\beta\). To an element \(b_\xi^{(\alpha)}\) of the base \(B_\alpha\) we put in correspondence an interval \(c_\xi^{(\alpha)}\) with ends from \(P_\alpha^\xi\) such that

\[ [H_{\alpha+1}]\subset c_\xi^{(\alpha)}\subset [c_\xi^{(\alpha)}]\subset H_\alpha . \]

As is easy to see, the family \(D_1\) of sets of the form \(b_\xi^{(\alpha)}\times c_\xi^{(\alpha)}\) forms a base at the points of the set \(\{\vartheta\}\times Y_1\) of the space \(Y_2=Y_1\times Y_1\). We note (this is important) that the intersection of the boundaries of two squares* of the family \(D_1\) is either empty, or consists of a finite number of points which are not vertices of squares of the family \(D_1\). To an element \(d=b_\xi^{(\alpha)}\times c_\xi^{(\alpha)}\) of the family \(D_1\) we put in correspondence (everywhere dense on the boundary) a set \(s(d)\) of boundary points at which at least one (and, as is easy to see, no more than one) coordinate belongs to \(Q_\alpha^\xi\). If a vertical or horizontal line intersects the boundary of a square of the family \(D_1\) in a whole interval, then it contains no more than two points of the set \(S_1=\bigcup_{d\in D_1}s(d)\). Let \(S_2\) be the set of those vertices of squares of the family \(D_1\) neither of whose coordinates is equal to \(\min Y_1\) or \(\max Y_1\). Let \(S_3\) consist of the points of the set \(\{\vartheta\}\times Y_1\) which do not lie on the boundaries of squares of the family \(D_1\).

Let \(D_2^*\) be the family of pairwise products of sets of the form \((r_1,r_2)\), or \([\min Y_1,r_2)\), or \((r_1,\max Y_1]\), where \(r_1,r_2\in R\). Let \(D_2\) be that part of \(D_2^*\) consisting of sets not intersecting \(\{\vartheta\}\times Y_1\). As is easy to see, the family \(D_2\) forms a base at the points of \(Y_2\setminus(\{\vartheta\}\times Y_1)\).

Let \(R^2=R\times R\).

A sequence \(\{\delta_\alpha\}_{\alpha\leq\omega_1}\), indexed by transfinite numbers not exceeding \(\omega_1\), of pairwise disjoint intervals lying in \(Y_1\), will be called marked if: a) \(\delta_\alpha\) consists of one point if \(\alpha\) is a limit transfinite number, and of more than one point if \(\alpha\) is an isolated transfinite number; b) the sequence \(\{\min\delta_\alpha\}\) is either decreasing or increasing; c) the set \(\bigcup_{\alpha\leq\omega_1}\delta_\alpha\) is closed in \(Y_1\). We assign to the family \(T\) a bicompactum \(t\) lying in \(Y_2\) if for it there is found a pair \(\{\delta_\alpha^1\}\) and \(\{\delta_\alpha^2\}\) of marked sequences (we shall then put one such pair in correspondence with the bicompactum \(t\)) such that \(t=\bigcup_{\alpha\leq\omega_1}t_\alpha\), where \(t_\alpha=t\cap(\delta_\alpha^1\times\delta_\alpha^2)\ne\varnothing\) and \(\operatorname{ind} t_\alpha=1\) if \(\alpha\) is an isolated transfinite number, and the single point from \(\delta_{\omega_1}^1\times\delta_{\omega_1}^2=t_{\omega_1}\subset t\) does not belong to \(R^2\).

Fix a decomposition of the set of all isolated transfinite numbers less than \(\omega_1\) into two sets \(\sigma_1\) and \(\sigma_2\), cofinal in \(\omega_1\).

There exists a mapping \(\lambda:K\to I\) of a Cantor perfect set onto an interval, which identifies pairwise the endpoints of adjacent intervals, takes them to all dyadic-rational points of the interval (except 0 and 1), and preserves order. We split the dyadic-rational points into two everywhere dense sets \(Q_1\) and \(Q_2\). Let \(\lambda_i\) \((i=1,2)\) be a mapping of the Cantor perfect set onto itself, identifying those points which are identified under \(\lambda\) and pass into \(Q_i\).

Let \(Z\) be the space of transfinite numbers not exceeding some transfinite number \(\alpha\). Between the transfinite numbers \(\beta\) and \(\beta+1\leq\alpha\) we insert (without making any identifications!) a copy \(K_\beta\) of the Cantor perfect set. The order in the resulting set \(C_\alpha\) is determined by the order on \(Z\) and on the Cantor perfect set and by the convention that all points of \(K_\beta\) lie between \(\beta\) and \(\beta+1\). The space \(C_\alpha\), taken in the order topolo-

* By a square we shall mean a set of the form \(J_1\times J_2\), where \(J_i\) \((i=1,2)\) is an interval, or a half-open interval, or a segment. By the boundary is meant the topological boundary with respect to \(Y_2\). A segment (horizontal or vertical, respectively) is a set of the form \(J\times\{y\}\) or \(\{y\}\times J\), where \(J\) is a (nontrivial) segment in \(Y_1\), \(y\in Y_1\). If \(J=Y_1\), then the corresponding segment is called horizontal or vertical.

is a bicompactum. The points of the set \(Z \subset C_\alpha\) will be called transfinites.

Let \(\chi\) be a transfinite whose character is greater than the cardinality of the set of all subsets of the space \(Y_2\). To each transfinite smaller than \(\chi\) we assign either some bicompactum of the family \(T\), or a point of the set \(Y_2 \setminus (R^2 \cup S_1)\), in such a way that the set of those transfinites to which a certain fixed point or a certain fixed bicompactum has been assigned is cofinal in \(\chi\).

In the product \(Y_2 \times C_\chi\) we make the following identifications:

a) if to a transfinite \(\beta < \chi\) there is assigned the bicompactum
\[ t=\bigcup_{\alpha \leqslant \omega_1} t_\alpha \]
of the family \(T\), then in the set \(\{y\} \times K_\beta\) we perform the identification \(\lambda\), if \(y \in t_\alpha\), where \(\alpha\) is a limit transfinite, and the identification \(\lambda_i\), if \(y \in t_\alpha\), where \(\alpha \in \sigma_i\);

b) if to a transfinite \(\beta < \chi\) there is assigned a point \(y_0 \in Y_2 \setminus (S_2 \cup S_3)\), then in the set \(\{y_0\} \times K_\beta\) we perform the identification \(\lambda\), in the set \(\{y\} \times K_\beta\) we perform the identification \(\lambda_1\), if \(y\) belongs to the vertical line passing through \(y_0\), and the identification \(\lambda_2\), if \(y\) belongs to the horizontal line passing through \(y_0\);

c) if to a transfinite \(\beta < \chi\) there is assigned a point \(y_0 \in S_2\), which is a vertex of a square \(d \in D_1\), then in the set \(\{y_0\} \times K_\beta\) we perform the identification \(\lambda\), in the set \(\{y\} \times K_\beta\), where \(y\) belongs to the horizontal or vertical line passing through the point \(y_0\), we perform the identification \(\lambda_1\), if the segment joining \(y\) to \(y_0\) intersects the boundary of the square in some (nontrivial) segment, and the identification \(\lambda_2\) otherwise;

d) if to a transfinite \(\beta < \chi\) there is assigned a point \(y_0 \in S_3\), then in the set \(\{y_0\} \times K_\beta\) we perform the identification \(\lambda\); in the set \(\{y\} \times K_\beta\), where \(y\) lies on the horizontal line passing through the point \(y_0\), we perform the identification \(\lambda_1\), if \(y\) lies to the left of \(y_0\), and the identification \(\lambda_2\), if \(y\) lies to the right of \(y_0\).

As a result of the factorization \(\Phi\) carried out, we obtain a certain (Hausdorff) bicompactum \(X\), with
\[ \operatorname{ind} X = 2,\qquad \operatorname{Ind} X = 3,\qquad X = X_1 \cup X_2, \]
where
\[ X_1=\Phi\bigl([\min Y_1,\vartheta]\times Y_1 \times C_\chi\bigr),\qquad X_2=\Phi\bigl([\vartheta,\min Y_1]\times Y_1 \times C_\chi\bigr), \]
\[ \operatorname{Ind} X_1=\operatorname{Ind} X_2=2. \]

We give an outline of the verification of the last assertions. First of all note that all dimensions of the spaces \(X, X_1, X_2\) are not less than 2 and not greater than 3.

Let \(V\) be a set open in \(Y_2\), and let
\[ Ц(V)=\Phi(V\times C_\chi). \]
Note that if \(V \in D_1 \cup D_2\), then the boundary of the set \(Ц(V)\) is one-dimensional. Let \(U\) be a set open in \(C_\chi\); the (open) set \(C_\lambda(U)\) consists of those points of the space \(X\) whose full inverse image under \(\Phi\) lies in \(Y_2 \times U\). As is easy to see, the sets of the form \(Ц(V)\cap C_\lambda(U)\) form a base of the space \(X\). At any point \(x=\Phi((y,c))\) there is an arbitrarily small neighborhood of this form with \(V \in D_1 \cup D_2\) such that, if \(c\) is a transfinite or the point \((y,c)\) belongs to a set \(\{y\}\times K_\beta\) in which the identification \(\lambda\) has not been performed, then the boundary of the neighborhood lies entirely in the boundary of \(Ц(V)\), or, if the point \((y,c)\) belongs to a set \(\{y\}\times K_\beta\) in which the identification \(\lambda\) has been performed, then the boundary of the neighborhood decomposes into the union of two closed one-dimensional sets \(F_1\) and \(F_2\), where \(F_1\) lies in the boundary of \(Ц(V)\), \(F_2\) lies in the boundary of \(C_\lambda(U)\), and either \(F_1 \cap F_2=\varnothing\), or \(F_1\) consists of a finite number of (nonintersecting) segments. In all these cases the boundary of the neighborhood is one-dimensional and, consequently, \(\operatorname{ind} X=2\). Every closed set in \(X_i\) (\(i=1\) or 2) has an arbitrarily tight neighborhood with one-dimensional boundary, which is the union of a finite number of sets of the form \(Ц(V)\cap C_\lambda(U)\cap X_i\), where \(V\in D_2^*\); therefore
\[ \operatorname{Ind} X_1=\operatorname{Ind} X_2=2. \]

Lemma 1. Let
\[ \Gamma_1=Y_1\times\{\min Y_1\},\qquad \Gamma_2=Y_1\times\{\max Y_1\}. \]
Let the set \(F\) separate \(\Gamma_1\) and \(\Gamma_2\). Then \(F\) contains either a) a bicompactum of the fam—

of \(T\), or b) a pair of intersecting segments, one of which lies on the boundary of some square of the family \(D_1\), while the other lies outside this boundary, or c) a horizontal segment not lying on the boundary of a square of the family \(D_1\), intersecting (passing from one side to the other) the segment \(\{\vartheta\}\times Y_1\), or d) a segment of the form \(\{\vartheta\}\times J\), or \(J\times\{y\}\), or \(\{y\}\times J\), where \(y\in Y_1\) is a point of countable character.

Lemma 2. Let \(G_1\) and \(G_2\) be two open disjoint sets in \(X\); \(F=X\setminus(G_1\cup G_2)\) consists of points limiting both for \(G_1\) and for \(G_2\). Then either \(F\) contains some set open in \(\Phi(Y_1\times\{\chi\})\), or there exists a transfinite \(\alpha<\chi\) such that, as soon as \(\Phi((y,\chi))\in F\) and \(\alpha\leq c\leq\chi\), then \(\Phi((y,c))\in F\).

From Lemmas 1 and 2 and the construction of the space \(X\) it follows easily that

\[ \operatorname{Ind} X=3. \]

I express my gratitude to A. V. Arhangel’skii, conversations with whom helped give the example its final form.

References

1. J. G. Ceder, Fund. Math., 55, No. 1, 87 (1964).