UDC 513.83

MATHEMATICS

V. KUZNETSOV

ON SPACES OF CLOSED SUBSETS

(Presented by Academician P. S. Aleksandrov on 28 IV 1967)

Let \(X\) be a topological space. Denote by \(\exp X\) the new topological space whose points are the nonempty closed subsets \(F\) of the space \(X\), and whose open base consists of the sets

\[ \langle U_1,\ldots,U_n\rangle = \mathscr{E}\left( F\in \exp X:\ F\subseteq \bigcup_{k=1}^{n} U_k,\ F\cap U_k\ne \varnothing,\ k=1,2,\ldots,n \right), \]

where \(U_1,\ldots,U_n\) is an arbitrary finite collection of nonempty open sets of the space \(X\). Numerous properties of the space \(\exp X\) are described in detail in \((^{2-5})\). It should be noted that the space \(\exp X\), endowed with the indicated topology, is always a \(T_0\)-space. If, however, \(X\) is normal, then \(\exp X\) is completely regular \((^2)\). Moreover, the conditions that \(X\) is (bi)compact and that \(\exp X\) is (bi)compact are equivalent \((^3)\).

In this note the connection between the weight characteristics of the spaces \(X\) and \(\exp X\) is investigated, a topological description of \(\exp X\) is given for certain spaces \(X\), and, finally, the connection between the dimensions of \(X\) and \(\exp X\) is studied.

Lemma 1. The system
\[ \mathfrak{B}=\{\langle U_{\alpha_1},\ldots,U_{\alpha_n}\rangle\}, \]
whose elements are all possible nonempty finite collections of sets of the open base
\[ \mathfrak{b}=\{U_\alpha\} \]
of the bicompactum \(X\), forms an open base of the bicompactum \(\exp X\).

Proof. Let \(F\in \langle G_1,\ldots,G_n\rangle \subseteq \exp X\). Since \(G_1,\ldots,G_n\) are open in \(X\), and \(\mathfrak{b}\) is a base in \(X\), we have

\[ G_k=\bigcup_{\xi} U_{\alpha(k)}^{\xi},\quad k=1,2,\ldots,n. \]

From the definition of the topology in \(\exp X\) it follows that in \(X\)

\[ F\subseteq \bigcup_{k=1}^{n}\ \bigcup_{\xi} U_{\alpha(k)}^{\xi}; \tag{1} \]

\[ F\cap G_k = F\cap\left(\bigcup_{\xi} U_{\alpha(k)}^{\xi}\right) = \bigcup_{\xi}\left(F\cap U_{\alpha(k)}^{\xi}\right)\ne \varnothing,\quad k=1,2,\ldots,n. \tag{2} \]

For each \(k=1,2,\ldots,n\) choose from the covering \(\{U_{\alpha(k)}^{\xi}\}\) of the bicompactum \(F\subseteq X\) those \(U_{\alpha(k)}^{\xi}\) for which \(F\cap U_{\alpha(k)}^{\xi}\ne \varnothing\). As (2) shows, such \(U_{\alpha(k)}^{\xi}\) exist for every \(k=1,2,\ldots,n\) and together form an open covering of \(F\). Choose from this covering a finite one. If, however, there is a \(k\), \(1\le k\le n\), such that not a single \(U_{\alpha(k)}^{\xi}\subseteq G_k\) participates in the obtained covering, then adjoin to the covering an arbitrary element \(U_{\alpha(k)}^{\xi}\) for which \(F\cap U_{\alpha(k)}^{\xi}\ne \varnothing\). We obtain a covering

\[ \{U_{\alpha(1)}^{1},\ldots,U_{\alpha(1)}^{p_1},\ldots,U_{\alpha(n)}^{1},\ldots,U_{\alpha(n)}^{p_n}\} \]

of the bicompactum \(F\), where

\[ U_{\alpha(k)}^{i}\subseteq G_k,\quad k=1,2,\ldots,n,\quad i=1,2,\ldots,p_k. \tag{3} \]

Obviously,

\[ F\in \langle U_{\alpha(1)}^{1},\ldots,U_{\alpha(1)}^{p_1},\ldots,U_{\alpha(n)}^{1},\ldots,U_{\alpha(n)}^{p_n}\rangle . \]

By virtue of (3) one obtains the inclusion

\[ \langle U_{\alpha(1)}^{1},\ldots,U_{\alpha(1)}^{p_1},\ldots,U_{\alpha(n)}^{1},\ldots,U_{\alpha(n)}^{p_n}\rangle \subseteq \langle G_1,\ldots,G_n\rangle . \]

The lemma is proved.

A system \(\sigma=\{U_\alpha\}\) of open sets of the space \(X\) is called dense (Ponomarev) in this space if, for every open \(G\subseteq X\), there exists \(U_\alpha\in\sigma\) such that \(U_\alpha\subseteq G\).

Lemma 2. The system \(\Sigma=\{\langle U_{\alpha_1},\ldots,U_{\alpha_n}\rangle\}\), whose elements are nonempty finite collections of sets from the system \(\sigma=\{U_\alpha\}\), dense in the space \(X\), is dense in \(\exp X\).

The least cardinal number that is the cardinality of an open base of the space \(X\) is called the weight of the space \(X\) and is denoted by \(wX\).

Theorem 1. The weight of the space \(\exp X\) coincides with the weight of \(X\), if \(X\) is a bicompactum and \(wX\geq \aleph_0\).

The least cardinal number that is the cardinality of a system of open sets dense in \(X\) is called the \(\pi\)-weight of the space \(X\) and is denoted by \(w_\pi X\) (Ponomarev).

Theorem 2. For an arbitrary topological space \(X\),
\[ w_\pi X=w_\pi(\exp X),\quad \text{if } w_\pi X\geq \aleph_0. \]

Let \(\mathfrak B_x=\{O_\alpha x\}\) be some fundamental system of neighborhoods of a point \(x\in X\). The cardinal number \(\inf_\alpha (wO_\alpha x)\) is called the local weight of the point \(x\) in the space \(X\).

The space \(X\) is called homogeneous with respect to local weight if \(w(x,X)\) is constant on \(X\) \((^6)\).

Theorem 3. The bicompactum \(\exp X\) is homogeneous with respect to local weight and \(w(F,\exp X)=\tau\) for every point \(F\in \exp X\), if \(X\) is homogeneous with respect to local weight and \(w(x,X)=wX=\tau\geq \aleph_0\).

Proof. A bicompactum \(X\) is homogeneous with respect to local weight and \(w(x,X)=wX=\tau\) if and only if for every point \(x\in X\) and any of its neighborhoods \(Ox\) there exists a bicompactum \(B_x\) such that \(x\in B_x\subseteq Ox\), \(wB_x=\tau\).

Take a point \(F\in \exp X\) and any of its neighborhoods \(\langle U_1,\ldots,U_n\rangle\) from the base of \(\exp X\). Since \(\exp X\) is a bicompactum, there exists a smaller neighborhood \(\langle V_1,\ldots,V_s\rangle\) of the point \(F\), for which
\[ F\in \langle V_1,\ldots,V_s\rangle\subseteq \langle \overline{V_1},\ldots,\overline{V_s}\rangle = \langle \overline{V}_1,\ldots,\overline{V}_s\rangle \subseteq \langle U_1,\ldots,U_n\rangle . \]
In each \(V_i\), \(i=1,2,\ldots,s\), there is contained a bicompactum \(B_i\) of weight \(\tau\); hence \(w\overline{V}_i=\tau\). To obtain a base \(\mathfrak B\) of the bicompactum \(\langle \overline{V}_1,\ldots,\overline{V}_s\rangle\), one must in each \(\exp \overline{V}_i\) choose a base \(\mathfrak B_i\) from sets of the form \(\langle U^i_{\alpha_1},\ldots,U^i_{\alpha_n}\rangle\), \(i=1,2,\ldots,s\), where \(\mathfrak b_i=\{U^i_\alpha\}\) is a base of \(\overline{V}_i\), and unite them over \(i=1,2,\ldots,s\) into common brackets:
\[ \langle U^1_{\alpha_1},\ldots,U^1_{\alpha_{p(1)}},\ldots,U^s_{\omega_1},\ldots,U^s_{\omega_{p(s)}}\rangle\in\mathfrak B . \]

If \(|\mathfrak B_i|=\tau\) *, \(i=1,2,\ldots,s\), then \(|\mathfrak B|=\tau\), and therefore
\[ w\langle \overline{V}_1,\ldots,\overline{V}_s\rangle\leq \tau . \]
Since all \(V_i\) are distinct, among them there will be some \(V_i\), and in it a bicompactum \(B_i\), such that \(B_i\cap V_j=\varnothing\) for \(i\ne j\), \(wB_i=\tau\). The bicompactum \(\exp B_i\) is topologically contained in \(\langle \overline{V}_1,\ldots,\overline{V}_s\rangle\). Since
\[ wB_i=w(\exp B_i)=\tau, \]
it follows that
\[ w\langle \overline{V}_1,\ldots,\overline{V}_s\rangle=\tau . \]
The theorem is proved.

The neighborhood character \(\chi(F,X)\) of a set \(F\) in the space \(X\) is the minimum of the cardinalities of fundamental systems of neighborhoods of \(F\) in \(X\) \((^7)\). The space \(X\) is called homogeneous with respect to neighborhood character if \(\chi(x,X)\) is constant on \(X\) \((^6)\). The pseudocharacter \(\psi(F,X)\) of a set \(F\) in \(X\) is the least number \(\mathfrak m\) such that
\[ F=\bigcap_{\alpha\in A} O_\alpha, \]
where all \(O_\alpha\) are open and \(|A|=\mathfrak m\) \((^7)\).

It is known that \(\psi(F,X)=\chi(F,X)\) if \(F\) is closed and \(X\) is a bicompactum.

Theorem 4. For an arbitrary topological space \(X\) and a closed \(F\subseteq X\),
\[ \chi(F,X)\leq \tau,\quad \text{if } \chi(F,\exp X)\leq \tau . \]

In \((^7)\) the following problem is formulated: does there exist a bicompactum with the first axiom of countability, having cardinality \(>\mathfrak c\), where \(\mathfrak c\) is the cardinality of the continuum?

Theorem 5. Any bicompactum \(Y=\exp X\) satisfying the first axiom of countability has cardinality \(\leq \mathfrak c\).

* The symbol \(|A|\) denotes the cardinality of the set \(A\).

Proof. Assuming the contrary, by means of Theorem 4 we conclude that the bicompactum \(X\) is perfectly normal*. In that case \(|X| \leq \mathfrak c\) (7). Since in \(X\) there are as many closed sets as open sets, and every open set, being of type \(F_\sigma\), is the sum of no more than a countable number of closed sets, it follows that \(|\exp X| \leq \mathfrak c\). The contradiction obtained proves the theorem.

A caliber of a space \(X\) is a cardinal number \(\mathfrak m\) such that every system of nonempty open sets of the space \(X\) of cardinality \(\mathfrak m\) has an equicardinal subsystem that has a nonempty intersection (Shanin).

It is known that every regular cardinal number \(\mathfrak m > \aleph_0\) is a caliber of any dyadic bicompactum (6).

Theorem 6. A regular cardinal number \(\mathfrak m > \aleph_0\) is a caliber of the space \(\exp X\) if and only if \(\mathfrak m\) is a caliber of \(X\).

Proof. We shall carry it out for a system of sets of an open base of the space \(\exp X\). Let
\[ \Sigma=\{\langle U_{\alpha_1},\ldots,U_{\alpha_s}\rangle\} \]
be a system of nonempty open sets in \(\exp X\) of regular cardinality \(\mathfrak m>\aleph_0\). It is assumed that the order of the elements in the brackets is fixed. The length of a set
\[ \langle U_{\alpha_1},\ldots,U_{\alpha_s}\rangle\in\Sigma \]
is the number \(s>0\). By virtue of the regularity of the number \(\mathfrak m\), there is a number \(n>0\) such that the system \(\Sigma\) contains an equicardinal subsystem \(\Sigma_n^{(0)}\), all elements of which have the same length \(n\). From each
\[ \langle U_{\alpha_1},\ldots,U_{\alpha_n}\rangle\in\Sigma_n^{(0)} \]
the first element \(U_{\alpha_1}\) is selected, and the system
\[ \sigma_1=\{U_{\alpha_1}\} \]
of the selected sets is considered in the space \(X\). Since \(\mathfrak m\) is a caliber of \(X\), there exists
\[ \sigma_1'\subseteq\sigma_1,\qquad |\sigma_1'|=|\sigma_1|=\mathfrak m, \]
having a nonempty intersection in \(X\). In \(\exp X\) we consider the system
\[ \Sigma_n^{(1)}\subseteq\Sigma_n^{(0)}, \]
for whose elements \(U_{\alpha_1}\in\sigma_1'\) in \(X\). With the system \(\Sigma_n^{(1)}\) the same is done as with \(\Sigma_n^{(0)}\), but with respect to the second elements \(U_{\alpha_2}\). As a result one obtains a system
\[ \Sigma_n^{(2)}\subseteq\Sigma_n^{(1)}. \]
Continuing the process described, at the \(n\)-th step we obtain a system
\[ \Sigma_n^{(n)}\subseteq\Sigma_n^{(n-1)}, \]
all \(n\)-th elements of whose sets belong to the system
\[ \sigma_n'=\{U_{\alpha_n}\}, \]
which has a nonempty intersection in \(X\). Thus:
\[ \Sigma \supseteq \Sigma_n^{(0)} \supseteq \Sigma_n^{(1)} \supseteq \ldots \supseteq \Sigma_n^{(n)}; \tag{4} \]
\[ \left|\Sigma_n^{(k)}\right|=|\Sigma|=\mathfrak m,\qquad k=0,1,\ldots,n; \tag{5} \]
\[ \Sigma_n^{(n)}=\{\langle U_{\alpha_1},\ldots,U_{\alpha_n}\rangle\},\qquad U_{\alpha_k}\in\sigma_k',\quad k=1,2,\ldots,n. \tag{6} \]

In the space \(X\), for each \(k=1,2,\ldots,n\), choose a point
\[ x_k\in\bigcap_{\sigma_k'} U_{\alpha_k}. \]
Then
\[ F=\{x_1,\ldots,x_n\}\in\exp X, \]
\[ F\in\bigcap_{\Sigma_n^{(n)}}\langle U_{\alpha_1},\ldots,U_{\alpha_n}\rangle. \]

Hence, and from (4), (5), the necessity of the condition follows. For the proof of sufficiency only the formula
\[ \bigcap_{\alpha}\langle U_\alpha\rangle=\left\langle\bigcap_{\alpha}U_\alpha\right\rangle \]
is needed (2).

A topological space \(X\) satisfies the Suslin condition if every system of disjoint open sets from \(X\) is at most countable (see (6)).

Corollary 1. The spaces \(X\) and \(\exp X\) simultaneously satisfy the Suslin condition if \(\aleph_1\) is a caliber of one of them.

Corollary 2. The space \(X\) satisfies the Suslin condition if the space \(\exp X\) satisfies it.

* A space \(X\) is called perfectly normal if it is normal and each of its closed sets is of type \(G_\delta\).

A topological space \(X\) satisfies the Knaster condition if every uncountable system of nonempty open subsets of \(X\) contains an uncountable subsystem every pair of whose sets has nonempty intersection (see (9)).

Theorem 7. The topological space \(\exp X\) satisfies the Knaster condition if and only if \(X\) satisfies this condition.

The density \(sX\) of a space \(X\) is the least cardinality of everywhere dense subsets of the space \(X\).

Theorem 8. The densities of the spaces \(X\) and \(\exp X\) coincide.

Denote by \(T_\tau\) the discrete \(T_1\)-space of cardinality \(\tau \geq \aleph_0\), and by \(\mathfrak b_0 T_\tau\) its one-point compactification with vertex \(\mathfrak b_0\) (the Alexandrov space) (see (1)).

Theorem 9. The bicompactum \(\exp \mathfrak b_0 T_\tau\) is a compactification of \(\mathfrak b T_\tau\) with remainder \(\mathfrak b T_\tau \setminus T_\tau\), homeomorphic to \(D^\tau\) for every \(\tau \geq \aleph_0\).

Proof. Denote
\[ X=\mathscr E(F\in \exp \mathfrak b_0T_\tau:\mathfrak b_0\notin F),\quad Y=\mathscr E(F\in \exp \mathfrak b_0T_\tau:\mathfrak b_0\in F). \]
Obviously, \(X\cup Y=\exp \mathfrak b_0T_\tau\), \(X\cap Y=\varnothing\), \(X\) is open, and \(Y\) is closed in \(\exp \mathfrak b_0T_\tau\). It is not hard to see that \(X\) is homeomorphic to \(T_\tau\). Renumber the points \(t\in T_\tau\):
\[ T_\tau=\{t_\alpha\}_{\alpha\in A}|A|=\tau \]
and represent
\[ D=\prod_{\alpha\in A} D_\alpha^{(0,1)}. \]
The mapping \(f:Y\to D^\tau\) consists in assigning to each point \(F\in Y\) the point \(x\in D^\tau\) with coordinates
\[ x_\alpha=1,\quad \text{if } t_\alpha\in F\subseteq \mathfrak b_0T_\tau,\qquad x_\alpha=0,\quad \text{if } t_\alpha\notin F. \]
It is easy to see that \(f\) is a homeomorphism onto all of \(D^\tau\).

Let \(\aleph_0 \leq \mathfrak m<\tau\). The \(\Sigma\mathfrak m\)-product is the set
\[ \Sigma D^\tau\subset D^\tau \]
consisting of those points \(x=\{x_\alpha\}\in D^\tau\) for which \(x_\alpha\ne 0\) for at most a set of indices of cardinality \(\leq \mathfrak m\) (6). Denote
\[ \exp_{\mathfrak m}X=\mathscr E(F\in \exp X:|F|\leq \mathfrak m) \]
for any cardinal number \(\mathfrak m\geq \aleph_0\).

Corollary. In \(\exp \mathfrak b_0T_\tau\),
\[ (\exp \mathfrak b_0T_\tau\setminus T_\tau)\cap \exp_{\mathfrak m}\mathfrak b_0T_\tau\cong \Sigma_{\mathfrak m}D^\tau. \]

The dimension of the space \(Y\) sometimes allows one to judge whether \(Y\) is representable in the form \(\exp X\).

Theorem 10. The compactum \(\exp X\) is infinite-dimensional if \(\dim X\geq 1\).

Proof. The mapping
\[ f_n:X^n\to \exp_n X \]
is (4) open-and-closed and \(n!\)-to-one. Therefore
\[ \dim X^n=\dim(\exp_n X). \]
Using results from (8), it is easy to obtain the inequality, for any \(n<\aleph_0\):
\[ \dim(\exp_n X)=\dim X^n\geq n,\quad \text{if } \dim X\geq 1. \]
Since \(\exp_n X\) is closed in \(\exp X\) for every \(n=1,2,\ldots\) and \(\dim(\exp_n X)\geq n\), the space \(\exp X\) is an infinite-dimensional compactum.

Corollary. The space
\[ \exp_{\aleph}X=\mathscr E(F\in \exp X:|F|<\aleph_0) \]
of all finite subsets of the compactum \(X\) is weakly infinite-dimensional.

It remains an open question: will \(\exp X\) be a strongly infinite-dimensional compactum if \(\dim X\geq 1\)?

The author expresses gratitude to P. S. Aleksandrov for his attention to this work and to B. A. Efimov for discussion and a number of valuable comments.

Moscow State University
named after M. V. Lomonosov

Received
12 IV 1967

REFERENCES

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\[ {}^{*}\ D^\tau \]
is the generalized Cantor set, i.e. the Tikhonov product of \(\tau\) discrete two-point spaces, \(\tau\geq \aleph_0\).