UDC 513.831

MATHEMATICS

S. SIROTA

ON THE SPECTRAL REPRESENTATION OF SPACES OF CLOSED SUBSETS OF BICOMPACTA

(Presented by Academician P. S. Aleksandrov on XII 1, 1967)

The space of closed subsets of a given topological space \(X\) in the Vietoris topology is called the space \(\exp X\), whose points are the closed subsets \(H\) of the space \(X\), and whose basic open sets are all sets of the form \(\langle U_{\alpha(1)}, \ldots\)

\[ \ldots, U_{\alpha(n)}\rangle = \left\{H=[H]\subset X:\quad H\subset \bigcup_{i=1}^{n} U_{\alpha(i)},\quad H\cap U_{\alpha(i)}\ne \varnothing,\quad i=1,\ldots,n \right\}, \]

where \(U_{\alpha(1)}, \ldots, U_{\alpha(n)}\) are open subsets of the space \(X\). As is known \((^2)\), \(\exp X\) is bicompact if and only if \(X\) is bicompact.

The main result of the paper is Theorem 1, which gives a spectral representation for the space of closed subsets of the limit of an inverse topological spectrum, and Theorem 3, in which the dyadicity of the space of closed subsets of a bicompactum is proved in the case when it itself is dyadic and its weight does not exceed \(\aleph_1\).

In what follows, the point of the space \(\exp X\) corresponding to a closed subset \(H\) of the space \(X\) will be denoted by the symbol \(\widehat H\).

Lemma 1. If \(\varphi: X\to Y\) is a continuous mapping of the space \(X\) into the space \(Y\), then the mapping \(\psi: \exp X\to \exp Y\), defined as follows: \(\psi(\widehat H)=\widehat{\varphi(H)}\), is a continuous mapping; moreover, \(\psi(\exp X)=\exp \varphi(X)\), and, if the mapping \(\varphi\) is open, then the mapping \(\psi\) is also open.

In what follows, the mapping \(\psi\) generated by the mapping \(\varphi\) by means of the construction described above will be called the exponential continuation of the mapping \(\varphi\).

Theorem 1. If \(\{X_\alpha;\pi_\beta^\alpha\}\) is a spectrum, where the \(X_\alpha\) are bicompacta, then \(\{\exp X_\alpha;\omega_\beta^\alpha\}\) is also a spectrum, and moreover

\[ Y=\lim_{\leftarrow}\{\exp X_\alpha;\omega_\beta^\alpha\} = \exp \lim_{\leftarrow}\{X_\alpha;\pi_\beta^\alpha\}, \]

where \(\omega_\beta^\alpha\) is the exponential continuation of the mapping \(\pi_\beta^\alpha\).

Proof. We shall first prove that \(\{\exp X_\alpha;\omega_\beta^\alpha\}\) is a spectrum. The set \(\{\alpha\}\) is directed, and it remains to show that if \(\alpha>\beta>\gamma\), then \(\omega_\gamma^\beta\omega_\beta^\alpha=\omega_\gamma^\alpha\). But for every \(H=[H]\subset X\), evidently,

\[ \omega_\gamma^\beta\omega_\beta^\alpha(\widehat H) = \omega_\gamma^\beta\widehat{\pi_\beta^\alpha(H)} = \widehat{\pi_\gamma^\beta\pi_\beta^\alpha(H)} = \widehat{\pi_\gamma^\alpha(H)} = \omega_\gamma^\alpha(\widehat H), \]

whence the required assertion follows.

As is known \((^3)\), if for the bicompactum \(\exp X\) there exists a family \(\{\varphi_\alpha\}\) of its mappings onto the spaces \(\exp X_\alpha\), with \(\omega_\beta^\alpha\varphi_\alpha=\varphi_\beta\), then it generates a continuous mapping \(\varphi\) of the bicompactum \(\exp X\) onto \(Y\). Denote by \(\pi_\alpha\) the natural projections

\[ X=\lim_{\leftarrow}\{X_\alpha;\pi_\beta^\alpha\} \]

onto \(X_\alpha\), and let \(\varphi_\alpha:\exp X\to \exp X_\alpha\) be the exponential continuation of the mapping \(\pi_\alpha\). We now prove that \(\omega_\beta^\alpha\varphi_\alpha=\varphi_\beta\). For \(\alpha>\beta\) we have

\[ \omega_\beta^\alpha\varphi_\alpha(\widehat H) = \omega_\beta^\alpha\widehat{\pi_\alpha(H)} = \widehat{\pi_\beta^\alpha\pi_\alpha(H)} = \widehat{\pi_\beta(H)} = \varphi_\beta(\widehat H). \]

Thus, for the space \(\exp X\) we have obtained a family of mappings satisfying the required conditions, which generates a continuous mapping \(\varphi\) onto \(Y\),

and, by the bicompactness of \(\exp X\), to prove the homeomorphism it suffices to show that \(\varphi\) is a condensation. Let \(\hat H_1 \in \exp X\), \(\hat H_2 \in \exp X\), \(\hat H_1 \ne \hat H_2\). Obviously,
\[ H_i=\lim_{\longleftarrow}\{\pi_\alpha(H_i);\ \pi_\beta^\alpha|\pi_\alpha(H_i)\}, \]
where \(\pi_\beta^\alpha|\pi_\alpha(H_i)\) is the restriction of the mapping \(\pi_\beta^\alpha\) to the set \(\pi_\alpha(H_i)\), \(i=1,2\), and, since \(H_1 \ne H_2\), for some \(\alpha\) we shall have \(\pi_\alpha(H_1)\ne \pi_\alpha(H_2)\), or, what is the same, \(\varphi_\alpha(\hat H_1)\ne \varphi_\alpha(\hat H_2)\). But this means that \(\omega_\alpha\varphi(\hat H_1)\ne \omega_\alpha\varphi(\hat H_2)\), whence it follows that \(\varphi(\hat H_1)\ne \varphi(\hat H_2)\). Hence \(\varphi\) is a condensation and a homeomorphism. The theorem is proved.

Lemma 2. There exists a topological embedding of the Cantor discontinuum \(C\) into the interval \([0,1]\) of the real line such that for any three points \(x,y,z\) of \(C\), if \(x\ne y\), then always \(|x-z|\ne |x-y|\).

In what follows the Cantor discontinuum will always be regarded as a subset of the line, situated in the indicated way.

Let \(H\) be a closed subset of \(C\). Denote
\[ D_0(H)=\{x\in H:\ 2x\le \inf H+\sup H\} \]
and, correspondingly,
\[ D_1(H)=\{x\in H:\ 2x\ge \inf H+\sup H\}. \]
In view of the remark made above, obviously,
\[ D_0(H)\cap D_1(H)=\varnothing. \]

Lemma 3. In the Hausdorff metric \(^*\) the relation \(D_i(H)\) is continuous, \(i=0,1\), i.e. if \(\rho_h(H_1,H_2)\to 0\), then always \(\rho_h(D_i(H_1),D_i(H_2))\to 0\).

Denote by \(AC\) the set of all perfect subsets of \(C\), and we shall regard \(AC\) as a subset of \(\exp C\).

Lemma 4. There exists a continuous mapping \(f\) of the product \(C\times AC\) onto \(C\) such that: 1) \(f(C\times\{\hat H\})=H\), where \(\hat H\in AC\), and 2) the restriction of \(f\) to the set \(C\times\{\hat H\}\) is a homeomorphism.

Here we shall carry out the construction of the mapping \(f\), and, for the sake of simplicity, omit the proof of the properties required by the lemma. If \(H\) is a perfect subset of \(C\), then, obviously, \(D_i(H)\) is also perfect, and for each sequence of zeros and ones \(\{i(1),\ldots,i(n)\}\) one can uniquely define \(D_{i(1)\ldots i(n)}(H)\) by the rule
\[ D_{i(1)\ldots i(n)}(H)=D_{i(n)}\bigl(D_{i(1)\ldots i(n-1)}(H)\bigr). \]
It is not difficult to show that all sets of this form constitute a base of \(H\), and each point \(x\in H\) determines a sequence \(I(x,H)=\{i(j)\}_{j=1}^{\infty}\) of zeros and ones, uniquely determined by the relations
\[ D_{i(1)}(H)\supset D_{i(1)i(2)}(H)\supset\cdots\ni x, \]
and conversely, each sequence \(I=\{i(j)\}_{j=1}^{\infty}\) determines a point \(x(I,H)\) satisfying the equality \(I(x(I,H))=I\). The mapping \(f\) is defined in the following way:
\[ f(x,\hat H)=x(I(x,C),H). \]
With the aid of Lemma 3 it is proved that \(f\) satisfies the required conditions.

Definition. A mapping \(\varphi\) of a bicompactum \(X\) onto a bicompactum \(Y\) is called a \(d\)-mapping if \(\varphi\) is continuous and open and there exists a topological embedding of \(X\) into \(C\times Y\) such that \(\varphi=\pi_y|X\), where \(\pi_y|X\) is the restriction of the projection of the product to the set \(X\).

Lemma 5. If \(\varphi\) is a \(d\)-mapping of the bicompactum \(X\) onto the bicompactum \(Y\), and the sets \(\varphi^{-1}(y)\) are perfect for every \(y\in Y\), then there exists a homeomorphism \(g:C\times Y\to X\) such that \(\pi_y=\varphi g\).

Proof. By the definition of a \(d\)-mapping there exists a topological embedding of \(X\) into \(C\times Y\) such that \(\varphi=\pi_y|X\). Define \(g\) then as follows:
\[ g(c,y)=f(c,\pi_c\varphi^{-1}(y),y), \]
where \(f\) is a mapping satisfying the conditions of the preceding lemma. It is not difficult to see that \(g\) is one-to-one and, moreover, since \(g(c,y)\in X\), we have
\[ \varphi g(c,y)=\pi_y|X(g(c,y))=y=\pi_y(c,y). \]
Let us now prove the continuity of the mapping \(g\). Denote \(H=\pi_c\varphi^{-1}(y)\). Let \(V_\varepsilon\times W\) be some basic

\(^*\) Recall the definition of the Hausdorff metric \(\rho_h\): if \(H_1\) and \(H_2\) are closed subsets of a metric compactum \(X\), then
\[ \rho_h(H_1,H_2)= \max_{x_1\in H_1}\min_{x_2\in H_2}\rho(x_1,x_2) + \max_{x_1\in H_2}\min_{x_2\in H_1}\rho(x_1,x_2). \]
The topology induced by the Hausdorff metric in \(\exp X\), for compact \(X\), coincides with the Vietoris topology \((^2)\).

neighborhood of the point \(g(c,y)\). Then, by Lemma 4, one can choose a neighborhood \(\langle U_1,\ldots,U_n\rangle\) of the point \(\tilde H\) in \(\exp C\) and a neighborhood \(U\) of the point \(c\) in \(C\) such that as soon as \(\tilde H_1\in \langle U_1,\ldots,U_n\rangle\) and \(c_1\in U\), then
\[ \rho\bigl(f(c,\tilde H),f(c_1,\tilde H_1)\bigr)<\varepsilon . \]
From the definition of the neighborhood \(\langle U_1,\ldots,U_n\rangle\) it follows that \(H\subset \bigcup_{i=1}^{n}U_i\), \(H\cap U_i\ne\varnothing\), \(i=1,\ldots,n\). But then \(C\times U_i\cap \varphi^{-1}(y)\ne\varnothing\) and is open, and
\[ O_i=\varphi\bigl(S\times U_i\cap \varphi^{-1}(y)\bigr)\ni y \]
and is also open by virtue of the openness of the mapping \(\varphi\). On the other hand,
\[ O_0=Y\setminus \varphi\bigl(X\setminus (C\times \bigcup_{i=1}^{n}U_i)\bigr) \]
is also open and contains \(y\). Consider now the neighborhood \(U\times (O\cap W)\) of the point \((c,y)\), where \(O=\bigcap_{i=0}^{n}O_i\). Then, if \((c_1,y_1)\in U\times (O\cap W)\), then
\[ \pi_c\varphi^{-1}\varphi(c_1,y_1)=\pi_c\varphi^{-1}(y_1)\in \langle U_1,\ldots,U_n\rangle,\quad c_1\in U, \]
and, by the choice of the neighborhoods \(\langle U_1,\ldots,U_n\rangle\) and \(U\), we have
\[ \rho\bigl(f(c,\pi_c\varphi^{-1}(y)),\ f(c_1,\pi_c\varphi^{-1}(y_1))\bigr)<\varepsilon, \]
i.e.
\[ f(c_1,\pi_c\varphi^{-1}(y_1))\in V_\varepsilon,\quad y_1\in W, \]
and
\[ g(c_1,y_1)=\bigl(f(c_1,\pi_c\varphi^{-1}(y_1)),y_1\bigr)\in V_\varepsilon\times W, \]
which is what had to be proved.

Lemma 6. If \(\varphi\) is a \(d\)-mapping of a bicompactum \(X\) onto a bicompactum \(Y\), then there exists a continuous mapping \(\psi\) of the product \(C\times Y\) onto \(X\), with \(\pi_y=\varphi\psi\), and \(\psi\) open.

Proof. It is not difficult to see that the superposition of two \(d\)-mappings is again a \(d\)-mapping. Denoting by \(\pi_x\) the projection of the product \(C\times X\) onto the factor \(X\), we have that \(\varphi\pi_x\) is a \(d\)-mapping, and for each \(y\in Y\) the set \(\pi_x^{-1}\varphi^{-1}(y)\) is perfect. Then, by Lemma 5, there exists a homeomorphism, and hence an open mapping,
\[ g:C\times Y\to C\times X, \]
with \(\pi_y=\varphi\pi_x g\). But then \(\psi=\pi_xg\) is the required mapping, since \(\pi_y=\varphi\psi\), and \(\psi\) is open as a superposition of two open mappings. The lemma is proved.

Lemma 7. If \(\varphi\) is a \(d\)-mapping of a bicompactum \(X\) onto a bicompactum \(Y\) and \(\psi\) is a continuous mapping of a bicompactum \(Z\) onto \(Y\), then there exists a mapping \(g\) of the product \(C\times Z\) onto \(X\) such that \(\psi\pi_z=\varphi g\); moreover, if \(\psi\) is an open mapping, then \(g\) can also be made open, and if \(\varphi\) is a \(d\)-mapping with perfect preimages for each \(y\in Y\) and \(\psi\) is a homeomorphism, then \(g\) can also be made a homeomorphism.

The proof follows immediately from the two preceding lemmas.

Theorem 2. If \(\{\alpha\}\) is a well-ordered set of indices and \(\{X_\alpha,\pi_\beta^\alpha\}\) is an inverse spectrum such that: 1) \(X_1\) is a dyadic bicompactum; 2) for every limit \(\alpha\) one has
\[ X_\alpha=\lim_{\leftarrow}\{X_\beta;\pi_\gamma^\beta\}_{\beta<\alpha}; \]
3) \(\pi_\alpha^{\alpha+1}\) is a \(d\)-mapping for all \(\alpha\), then
\[ X=\lim_{\leftarrow}\{X_\alpha;\pi_\beta^\alpha\} \]
is a dyadic bicompactum.

Proof. To each index \(\alpha\) we associate its own copy \(C_\alpha\) of the Cantor discontinuum. Next, putting
\[ Y_\alpha=X_1\times \prod_{\beta<\alpha} C_\beta \]
and taking \(\omega_\beta^\alpha\) to be the natural projection of the product onto the subproduct, we may consider the transfinite spectrum \(\{Y_\alpha;\omega_\beta^\alpha\}\). Obviously,
\[ \omega_\gamma^\beta\omega_\beta^\alpha=\omega_\gamma^\alpha, \]
and for limit \(\alpha\) one has
\[ Y_\alpha=\lim_{\leftarrow}\{Y_\beta;\omega_\gamma^\beta\}_{\beta<\alpha}. \]
To prove the dyadicity of \(X\), it is enough to construct a continuous mapping
\[ Y=\lim_{\leftarrow}\{Y_\alpha;\omega_\beta^\alpha\}, \]
which is a dyadic bicompactum, onto \(X\), for which in turn it is enough to find a family of continuous mappings, commuting with the spectral ones, which, by known theorems \((^3)\), will generate a continuous mapping of the limit onto the limit. We shall construct the desired family by the method of transfinite induction. As \(\varphi_1:Y_1\to X_1\) take the identity mapping. Suppose now that for all \(\beta<\alpha\) mappings \(\varphi_\beta\) have been constructed, and for every \(\gamma<\beta<\alpha\) the relation
\[ \pi_\gamma^\beta\varphi_\beta=\varphi_\gamma\omega_\gamma^\beta \]
holds. Then, in the case when \(\alpha\) is a limit transfinite, earlier

the constructed mappings induce a continuous mapping of the limit, and, since by the hypothesis of the theorem
\(X_\alpha=\lim_{\leftarrow}\{X_\beta;\pi_\gamma^\beta\}_{\beta<\alpha}\) and, by construction,
\(Y_\alpha=\lim_{\leftarrow}\{Y_\beta;\omega_\gamma^\beta\}_{\beta<\alpha}\), we have thereby obtained the required mapping \(\varphi_\alpha\). If, however, \(\alpha\) is an indeterminate transfinite number, then \(\alpha=(\alpha-1)+1\), and by Lemma 7 there is a mapping \(\varphi_\alpha\) such that \(\pi_{\alpha-1}^{\alpha}\varphi_\alpha=\varphi_{\alpha-1}\omega_{\alpha-1}^{\alpha}\), and consequently \(\varphi_\alpha\) is the required one. Thus a family of mappings has been found which generates a mapping continuous from the limit onto the limit. The theorem is proved.

Similarly to Theorem 2, with the aid of Lemma 7 one proves

Theorem 3. If, in the hypotheses of Theorem 2, the bicompactum \(X_1\) is moreover an open image of \(D^\tau\), then the inverse limit of the spectrum \(\{X_\alpha;\pi_\beta^\alpha\}\) is an open image of the generalized Cantor discontinuum.

Theorem 4. If, in the hypotheses of Theorem 2, for every \(\alpha\) the mapping \(\pi_\alpha^{\alpha+1}\) exists and is, moreover, a mapping with perfect point-preimages, then the limit of the spectrum \(\{X_\alpha;\pi_\beta^\alpha\}\) is homeomorphic to the topological product \(X_1\) by \(D^\tau\).

It is clear that, if the hypotheses of Theorem 4 are satisfied and \(X_1\) is a zero-dimensional compactum or the generalized Cantor discontinuum, then the limit of the spectrum \(\{X_\alpha;\pi_\beta^\alpha\}\) is homeomorphic to \(D^\tau\).

Corollary 1. If \(X\) is the limit of a transfinite spectrum of zero-dimensional compacta with open mappings, then \(X\) is an open image of \(D^\tau\).

Theorem 5. If \(X\) is a dyadic bicompactum of weight not exceeding \(\aleph_1\), then the bicompactum \(\exp X\) is dyadic.

Proof. As is easy to see, for the proof of the theorem it is enough to show that \(\exp D^{\aleph_1}\) is homeomorphic to \(D^{\aleph_1}\). Indeed, by virtue of the dyadicity of \(X\) there exists a continuous mapping of \(D^{\aleph_1}\) onto \(X\). Then, by Lemma 1, there exists a continuous mapping of \(\exp D^{\aleph_1}\) onto \(\exp X\), and if \(\exp D^{\aleph_1}\) is a dyadic bicompactum, then the theorem will be proved. Let \(\{\alpha\}\) be the set of transfinite numbers less than \(\omega_1\). To each \(\alpha\) assign a copy \(C_\alpha\) of the Cantor discontinuum and put
\(X_\alpha=\prod_{\beta<\alpha} C_\beta\). Denoting by \(\pi_\beta^\alpha\) the natural projection of the product onto a subproduct, we have \(\pi_\gamma^\alpha\pi_\beta^\gamma=\pi_\beta^\alpha\), and then \(\{X_\alpha;\pi_\beta^\alpha\}\) is an inverse transfinite spectrum whose limit, obviously, is \(D^{\aleph_1}\). But the space of closed subsets of a metric compactum is again a metric compactum, and, by Theorem 1, \(\exp D^{\aleph_1}\) is the inverse limit of a transfinite spectrum of compacta with mappings open by Lemma 1. Moreover, it is not hard to see that the exponential extension of the mapping \(\pi_\beta^\alpha\) is a mapping with perfect point-preimages, and, by Theorem 4, in view of Corollary 1, there follows a homeomorphism of \(\exp D^{\aleph_1}\) and \(D^{\aleph_1}\). The theorem is proved.

With the application of Lemma 1 one also proves

Corollary 2. If \(X\) is an open image of \(D^{\aleph_1}\), then \(\exp X\) is an open image of \(D^{\aleph_1}\).

The author expresses gratitude to P. S. Aleksandrov for the attention he has shown, and to B. A. Efimov for valuable suggestions.

Moscow Geological Prospecting Institute
named after S. Ordzhonikidze

Received
17 XI 1967

REFERENCES CITED

1. P. S. Aleksandrov, Introduction to the Theory of Sets and Functions, Moscow—Leningrad, 1948.
2. K. Kuratowski, Topology, 1, Moscow, 1966.
3. N. Steenrod, S. Eilenberg, Foundations of Algebraic Topology, Moscow, 1958.