UDC 513.83

MATHEMATICS

V. L. KLYUSHIN

ON SPACES DECOMPOSABLE INTO SPECTRA CONSISTING OF METRIC SPACES

(Presented by Academician P. S. Aleksandrov on 9 VI 1965)

In the present paper we consider spaces that are limits of inverse spectra of metric spaces. Such spaces were considered earlier by B. A. Pasynkov in his work (¹), where, in particular, necessary and sufficient conditions are given for the spectral decomposability of spaces with respect to the class of metric spaces. Theorem 1 establishes a one-to-one correspondence between the completion of a uniform space with respect to its given structure \(\Sigma\) and an inverse spectrum of complete metric spaces with uniformly continuous projections, called extremal. Next, the connection between spaces complete in the sense of Dieudonné and spectra of metric spaces is considered without using the concept of uniformity. Theorem 3 establishes, for finite-dimensional uniform spaces, the existence of spectra consisting of metric spaces of the same dimension.

By a spectrum here we shall always mean a directed set of spaces \(X_\alpha\), connected with one another by continuous mappings—projections
\(\mathfrak{D}^{\beta}_{\alpha}: X_\beta \to X_\alpha\), defined whenever \(\beta > \alpha\), and satisfying the transitivity condition:
\(\mathfrak{D}^{\gamma}_{\alpha}=\mathfrak{D}^{\beta}_{\alpha}\mathfrak{D}^{\gamma}_{\beta}\) for \(\gamma > \beta > \alpha\).

B. A. Pasynkov in (²) introduced the notion of an extremal spectrum for bicompacta and established a one-to-one correspondence between bicompacta and their extremal spectra. We define the notion of an extremal spectrum consisting of metric spaces.

Definition 1. Two spaces \(X_\alpha\) and \(X_\beta\) of a spectrum \(S\) will be called \(S\)-homeomorphic if there exists such a homeomorphism \(X_\beta \to X_\alpha\) that, under \(x_\beta \to x_\alpha\), the conditions \(\mathfrak{D}^{\gamma}_{\alpha}x_\gamma=x_\alpha\) and \(\mathfrak{D}^{\gamma}_{\beta}x_\gamma=x_\beta\) are equivalent for every \(\gamma>\alpha,\beta\). If this homeomorphism is uniform in both directions, then these two spaces will be called \(S\)-isomorphic.

Definition 2. A subspectrum \(S'\) of a spectrum \(S\) is called splitting if, for any \(X_{\alpha_0}\) from \(S\) and for any two points \(x'_{\alpha_0}\) and \(x''_{\alpha_0}\) in \(X_{\alpha_0}\), there exists such a \(\beta>\alpha_0\) that, for any two points \(x'_\beta\) and \(x''_\beta\) satisfying the conditions
\(\mathfrak{D}^{\beta}_{\alpha_0}x'_\beta=x'_{\alpha_0}\),
\(\mathfrak{D}^{\beta}_{\alpha_0}x''_\beta=x''_{\alpha_0}\), there is in \(S'\) a space \(X_\gamma\) in which
\(\mathfrak{D}^{\beta}_{\gamma}x'_\beta \ne \mathfrak{D}^{\beta}_{\gamma}x''_\beta\).

Definition 3. We shall say that the order in the spectrum
\[ S=\{X_\alpha,\mathfrak{D}^{\beta}_{\alpha}\} \]
is maximal if there exists no spectrum \(S'=\{X_\gamma,\mathfrak{D}^{\gamma}\}\), distinct from it, consisting of the same elements \(X_\alpha\), and such that from \(\beta>\alpha\) in \(S'\) it follows that \(\beta>\alpha\) in \(S\).

Definition 4. A spectrum
\[ S=\{X_\alpha,\mathfrak{D}^{\beta}_{\alpha}\} \]
is called extremal if it satisfies the following conditions: 1) it contains no \(S\)-homeomorphic spaces; 2) it has maximal order; 3) it is not

a splitting subspectrum of any spectrum satisfying conditions 1) and 2).

Definition 5. We shall call a spectrum \(S=\{X_\alpha,\ \mathfrak D_\alpha^\beta\}\) uniform if all projections \(\mathfrak D_\alpha^\beta\) are uniformly continuous.

Definition 6. A uniform spectrum \(S=\{X_\alpha,\ \mathfrak D_\alpha^\beta\}\) is called extremal if it satisfies the following conditions: 1) it contains no \(S\)-isomorphic spaces; 2) it has a maximal order; 3) it is not a splitting subspectrum of any uniform spectrum satisfying conditions 1) and 2).

Definition 7. Two spectra \(S=\{X_\alpha,\ \mathfrak D_\alpha^\beta\}\) and \(S'=\{X_\gamma,\ \mathfrak D_\gamma^\nu\}\) will be called equivalent (isomorphic) if there exists an isomorphism between the sets of their indices and, for each pair of corresponding indices, a homeomorphism (a homeomorphism uniform in both directions) \(X_\alpha\leftrightarrow X_\gamma\).

Theorem 1. The completion of a uniform space \(X\) with respect to its structure \(\Sigma\) is the limit of a unique, up to isomorphism, uniform extremal spectrum \(S=\{X_\alpha,\ \mathfrak D_\alpha^\beta\}\) of complete metric spaces, and the structure \(\Sigma\) is determined by the preimages of all metric covers of the elements of the spectrum \(S\). Therefore there is a one-to-one correspondence between complete uniform spaces and their extremal spectra.

Proof. Consider all possible uniformly continuous mappings \(\mathfrak D_\alpha:X\to X_\alpha\) of the space \(X\) onto metric spaces. Take the completions \(\widetilde X_\alpha\) of all these metric spaces \(X_\alpha\). We obtain a system of pairs \((\mathfrak D_\alpha,\widetilde X_\alpha)\), where \(\widetilde X_\alpha\) are complete metric spaces and \(\widetilde{\mathfrak D}_\alpha\) are uniformly continuous mappings. We shall regard the pairs \((\mathfrak D_\xi,\widetilde X_\xi)\) and \((\mathfrak D_\eta,\widetilde X_\eta)\) as isomorphic if there exists a homeomorphism, uniform in both directions, \(\widetilde X_\xi\to\widetilde X_\eta\) such that \(\mathfrak D_\xi=\mathfrak D_\xi^\eta\mathfrak D_\eta\). Divide the whole system into classes of \(S\)-isomorphic pairs and choose one representative from each class. The system of pairs thus obtained is directed; for any two pairs \((\mathfrak D_\xi,\widetilde X_\xi)\) and \((\mathfrak D_\eta,\widetilde X_\eta)\) one can take the pair \((\mathfrak D_\zeta,\widetilde X_\zeta)\), where

\[ \mathfrak D_\zeta(x)=(\mathfrak D_\xi(x),\ \mathfrak D_\eta(x))\in \widetilde X_\xi\times\widetilde X_\eta, \]

\[ X_\zeta=\mathfrak D_\zeta(X)\subset \widetilde X_\xi\times\widetilde X_\eta, \]

\(\widetilde X_\zeta\) is the completion of \(X_\zeta\), and \(\widetilde{\mathfrak D}_\zeta\) is the extension of \(\mathfrak D_\zeta\). We introduce an order: put \(\beta>\alpha\) if there exists a uniformly continuous projection \(\mathfrak D_\alpha^\beta\) satisfying the condition \(\mathfrak D_\alpha=\mathfrak D_\alpha^\beta\mathfrak D_\beta\). Since in the system of pairs there are no \(S\)-isomorphic ones, it cannot happen simultaneously that \(\beta>\alpha\) and \(\alpha>\beta\). From the system of pairs obtained we construct a spectrum \(S=\{X_\alpha,\ \mathfrak D_\alpha^\beta\}\). The limit of this spectrum will be the completion \(\widetilde X\) of the space \(X\) with respect to the structure \(\Sigma\). Indeed, first, for every \(\alpha\), \(X_\alpha=\mathfrak D_\alpha(X)\); consequently \(X\) is everywhere dense in the limit of the spectrum \(S\). Secondly, the structure \(\Sigma\) consists of the preimages of all metric covers of the spaces \(X_\alpha\). Our uniform spectrum \(S\) is extremal. The fulfillment of conditions 1) and 2) of Definition 6 follows from the very construction of the spectrum \(S\). Suppose that condition 3) of this definition is not fulfilled, i.e. the spectrum \(S\) is a splitting subspectrum of another uniform spectrum \(S^*\), not coinciding with it, of complete metric spaces. But then there exists a space \(X_{\alpha^*}\) belonging to \(S^*\) and not belonging to \(S\). Since the spectra \(S\) and \(S^*\) have one and the same limit, \(S^*\) also consists of the completions of all uniformly continuous images of \(\widetilde X\) and, by the supposition, contains \(S^*\)-isomorphic spaces, which contradicts condition 1).

Let us now verify the uniqueness of the extremal spectrum. Suppose there is a uniform extremal spectrum \(S^*\), consisting of complete metric spaces and having as its limit the space \(\widetilde X\). We shall show—

namely, that it is a subspectrum of the spectrum \(S\). Indeed, all mappings
\(\tilde{\omega}_\nu:\tilde{X}\to\tilde{X}_\nu\in S^*\) are uniformly continuous with respect to the structure \(\Sigma\), induced by the metric coverings of the elements of the spectrum \(S^*\). Hence all elements of the spectrum \(S^*\) are completions of uniformly continuous images of \(X\). Since among all pairs \((\tilde{\omega},\tilde{X}_\nu)\) from \(S^*\) there are no \(S^*\)-isomorphic ones, the spectrum \(S^*\) is a subspectrum of the spectrum \(S\). Moreover, the spectra \(S\) and \(S^*\) have one and the same limit. Therefore \(S^*\) is a refining subspectrum of the spectrum \(S\) and, consequently, coincides with it.

Theorem 2. Every complete uniform space in the sense of Diedonné is the limit of a unique, up to equivalence, extremal spectrum of metric spaces.

Proof. Consider all possible pairs \((\tilde{\omega}_\alpha,X_\alpha)\), where \(X_\alpha\) are metric spaces, and \(\tilde{\omega}_\alpha\) are continuous mappings \(\tilde{\omega}_\alpha:X\to X_\alpha\) onto these metric spaces. We introduce an order in the following way: \(\beta>\alpha\), if there exists a mapping \(\tilde{\omega}_\alpha^\beta:X_\beta\to X_\alpha\) satisfying the condition \(\tilde{\omega}_\alpha=\tilde{\omega}_\alpha^\beta\tilde{\omega}_\beta\). Repeating the arguments of B. A. Pasynkov, carried out in (1) in constructing a completion of a space with the maximal structure, we obtain a spectrum
\(S=\{X_\alpha,\tilde{\omega}_\alpha^\beta\}\), whose limit is the space \(X\) and which satisfies conditions 1) and 2) of Definition 4. The proof of the fact that \(S\) is not a refining subspectrum of any other spectrum \(S^*\) satisfying conditions 1) and 2) is carried out exactly as in the preceding theorem: assuming the contrary, there exists in \(S^*\) a space \(X_{\alpha^*}\) not belonging to \(S\), and since \(S\) contains all pairs \((\tilde{\omega}_\alpha,X_\alpha)\), there must exist \(S^*\)-homeomorphic spaces in \(S^*\). Uniqueness is proved by an almost literal repetition of the arguments of Theorem 1.

It is also not difficult to establish a one-to-one correspondence, up to equivalence, between complete spaces in the sense of Diedonné and extremal spectra of complete metric spaces.

Let us now consider finite-dimensional uniform spaces. We shall say, following Isbell, that a uniform space \(X\) has uniform dimension \(\Delta dX=n\), if for every covering \(\omega\) from its structure \(\Sigma\) there exists an \((n+1)\)-fold covering \(\eta\in\Sigma\) inscribed in \(\omega\).

The metric dimension \(\mu dM\) of a metric space \(M\) is the least number \(n\) such that for every \(\varepsilon>0\) there exists an open covering of the space \(M\) of multiplicity \(\leq n+1\), each element of which has diameter less than \(\varepsilon\).

Theorem 3. If a complete uniform space \(X\) has uniform dimension \(\Delta dX\leq n\), then in the extremal spectrum \(S=\{X_\alpha,\tilde{\omega}_\alpha^\beta\}\) of the space \(X\) there is a cofinal subspectrum \(S'=\{X_{\alpha'},\tilde{\omega}_{\alpha'}^\beta\}\) consisting of metric spaces having metric dimension \(\mu dX_{\alpha'}\leq n\).

Proof. Take an arbitrary covering \(\omega_0\) from the given structure \(\Sigma\) of the space \(X\). Since \(\Delta dX\leq n\), there exists a normal sequence \(\{\omega_i\}\) of coverings of multiplicity \(\leq n+1\), inscribed in \(\omega_0\) and belonging to the structure \(\Sigma\). Take the metric space \(X_{\omega_0}\) induced by this sequence. We shall show that \(\mu dX_{\omega_0}\leq n\). Indeed, let \(\varepsilon=1/2^k\) be given. Consider \(\omega_{k+3}\). We shall assume that the set of elements of the covering \(\omega_{k+2}\) is well ordered, \(\omega_{k+2}=\{U_\lambda\}_{\lambda\in\Lambda}\). Denote by \(O_{\omega_{k+3}}A\) the star of the set \(A\) with respect to the covering \(\omega_{k+3}\),
\(O_{\omega_{k+3}}A=\bigcup\{U;\ U\cap A\ne\Lambda,\ U\in\omega_{k+3}\}\). The system of open sets
\(\{f(O_{\omega_{k+3}}x)\}_{x\in X}\) is a \(1/2^{k+1}\)-covering of \(X_{\omega_0}\). Put

\[ O_\lambda=\bigcup\{O_{\omega_{k+3}}x;\ O_{\omega_{k+3}}x\subseteq U_\lambda\}. \]

The system \(\{O_\lambda\}_{\lambda\in\Lambda}\) is a covering of the space \(X\) of multiplicity not exceeding \(n+1\). For each \(\lambda\) the set \(f(O_\lambda)\) is open in \(X_{\omega_0}\), and its diameter is less than \(\varepsilon\). Thus, for every covering \(\omega\in\Sigma\) there exists a uniform \(\omega\)-mapping onto a metric space \(X_\omega\) of dimension \(\mu d X_\omega\le n\). Take the system of pairs \((\mathfrak{F}_\alpha,X_\alpha)\), where the \(X_\alpha\) are metric spaces, \(\mu d X_\alpha\le n\), and the \(\mathfrak{F}_\alpha\) are uniformly continuous mappings. Setting \(\beta' > \alpha'\) whenever there exists a uniformly continuous mapping
\(\mathfrak{F}_{\alpha'}^{\beta'}: X_{\beta'}\to X_{\alpha'}\), we obtain a spectrum \(S'=\{X_{\alpha'},\mathfrak{F}_{\alpha'}^{\beta'}\}\) over a directed set, which is a cofinal subspectrum of the extremal spectrum \(S\).

In conclusion I express my gratitude to V. I. Ponomarev and B. A. Pasynkov for their attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
8 VI 1965

REFERENCES

\(^1\) B. A. Pasynkov, Matem. sborn., 66, No. 1, 35 (1965).
\(^2\) B. A. Pasynkov, Tr. Tbilissk. matem. inst., 27, 43 (1960).
\(^3\) B. A. Pasynkov, DAN, 131, No. 2 (1960).
\(^4\) J. Isbell, Pacific J. Math., 9, No. 1 (1959).