MATHEMATICS

E. G. SKLYARENKO

ON THE REPRESENTATION OF INFINITE-DIMENSIONAL COMPACTA AS AN INVERSE LIMIT OF POLYHEDRA

(Presented by Academician P. S. Aleksandrov, 17 V 1960)

Freudenthal proved \((^1)\) that every compactum \(X\) can be represented as such an inverse limit of a sequence of polyhedra \(\{P_i, f_i^j\}\), in which the mappings \(f_i^j\) are piecewise-affine irreducible\(^*\) mappings “onto.” Moreover, in order that the compactum be \(n\)-dimensional, it is necessary and sufficient that it be representable as an inverse limit of a sequence of \(n\)-dimensional polyhedra in which the mappings \(f_i^j\) satisfy the conditions stated above.

The aim of the present note is to give an analogous characterization for one class of infinite-dimensional compacta, namely compacta that are the sum of a countable number of their closed finite-dimensional subsets. For brevity we shall call such compacta weakly countable-dimensional.\(^ {**}\)

Theorem 1. In order that a compactum \(X\) be weakly countable-dimensional, it is necessary and sufficient that it can be represented as an inverse limit of a sequence of polyhedra \(\{P_i, f_i^j\}\) such that the mappings \(f_i^j\) satisfy Freudenthal’s conditions and, for every thread \(\xi=\{\xi_i\}\), \(\xi_i \in P_i\), the dimensions of the carriers \(T(\xi_i)\) are bounded in the aggregate.

In proving this theorem we shall, in the main outlines, follow Freudenthal.

The proof of necessity is based on the following proposition: the space \(X\) is weakly countable-dimensional if and only if there exists a sequence of starwise inscribed in one another, shrinking\(^ {***}\), finite open coverings such that at each point \(x \in X\) the orders of all the coverings are bounded in the aggregate \((^{3,4})\). Let \(\{\alpha_n\}\) be a system of such coverings of the compactum \(X\). We shall construct a sequence of polyhedra \(P_i\) and mappings \(g_i : X \to P_i\), and also

\(^*\) A mapping is called irreducible if, after an admissible deformation, it remains a mapping “onto.” Here a deformation of a mapping is called admissible if, in the process of deformation, the image of each point does not leave its carrier. Finally, the carrier of a point \(\xi\) in a given complex is the closed simplex of least dimension containing this point; we shall denote it by \(T(\xi)\). Irreducibility of a mapping is equivalent to the fact that, for every closed simplex, its full preimage is mapped onto it essentially.

\(^ {**}\) Countable-dimensional spaces are those that are countable sums of their zero-dimensional (not necessarily closed) subsets. An example of a countable-dimensional but not weakly countable-dimensional compactum was constructed by Yu. M. Smirnov \((^2)\).

\(^ {***}\) A covering \(\beta\) is starwise inscribed in a covering \(\alpha\) if the star of any element of \(\beta\) with respect to \(\beta\) is contained in some element of \(\alpha\). The star of a set with respect to a covering is the sum of the elements of the covering that intersect the set. A sequence of coverings is called shrinking if, for every point and every neighborhood of it, there is a covering in the sequence such that the star of the point with respect to it is contained in the given neighborhood.

maps \(f_i^j:P_j\to P_i\) such that the following conditions are satisfied:
1) \(\dim T(g_i x)\leq k(x)\) for every point \(x\in X\), where \(k(x)\) is the upper bound of the multiplicities of the coverings \(\alpha_n\) at the point \(x\); 2) the mappings \(g_i\) are irreducible mappings “onto”; 3) the mappings \(f_i^j\) are piecewise affine; 4) \(T(f_i^j,g_j x)\subseteq T(g_i x)\) for each point \(x\in X\); 5) for every simplex \(T\in P_j\) the diameter of its image \(f_i^j T\) does not exceed \(1/2^{j-1}\).

For the construction of the polyhedron \(P_1\) and the mapping \(g_1\) we proceed as follows: first we construct the barycentric mapping of the compactum \(X\) into the nerve of the covering \(\alpha_1\), and then, applying the sweeping-out operation \((^5)\), we obtain an irreducible mapping \(g_1\) onto some subcomplex \(P_1\) of the nerve. If the polyhedra \(P_1,\ldots,P_r\) and the mappings \(g_i, f_i^j\), \(i\leq j\leq r\), satisfying the indicated conditions have already been constructed, then the construction of the polyhedron \(P_{r+1}\) and the mapping \(g_{r+1}\) is carried out as follows: we choose a sufficiently fine subdivision of the polyhedron \(P_r\) so that the diameters of the images of the simplexes of this subdivision under the mappings \(f_i^r\) do not exceed \(1/2^r\). By compactness of the space \(X\), in the sequence of coverings \(\{\alpha_n\}\) there is a covering, say \(\alpha_{r+1}\), inscribed in the covering composed of the inverse images of the principal stars of the subdivision of the polyhedron \(P_r\) under the mapping \(g_r\). Next, the polyhedron \(P_{r+1}\) and the mapping \(g_{r+1}\) are constructed from the covering \(\alpha_{r+1}\) in exactly the same way as the polyhedron \(P_1\) and the mapping \(g_1\) were constructed from the covering \(\alpha_1\); in this process, as the mapping \(f_r^{r+1}\) we take the simplicial mapping of the polyhedron \(P_{r+1}\) into the subdivision of the polyhedron \(P_r\), and, finally, put \(f_i^{r+1}=f_i^r f_r^{r+1}\). It is easy to verify that the sequence obtained in this way satisfies all the conditions formulated above.

We shall show that the inverse-limit compactum \(P\) of the sequence \(\{P_i,f_i^j\}\) is homeomorphic to \(X\). To this end we construct mappings \(f_i:X\to P_i\) such that \(f_i=f_i^j f_j\). For each mapping \(f_i\) we take the limit of the sequence of mappings \(f_i^j g_j\) (\(i\) fixed), which converges uniformly, since

\[ \rho\bigl(f_i^{j+1}g_{j+1}x,f_i^j g_jx\bigr) =\rho\bigl(f_i^j f_j^{j+1}g_{j+1}x,f_i^j g_jx\bigr) \leq 1/2^{j-1}, \]

which in turn follows from the fact that \(g_jx\) and \(f_j^{j+1}g_{j+1}x\) belong to one and the same closed simplex. Since the mapping \(f_i\) can be obtained from \(g_i\) by means of an admissible deformation, \(f_i\) is an irreducible mapping onto the polyhedron \(P_i\). Assigning to each point \(x\in X\) the thread \(\{f_i x\}\), we obtain the desired homeomorphism of the compactum \(X\) onto \(P\).

It remains to verify that for each thread \(\xi=\{\xi_i\}\), \(\xi_i\in P_i\), the dimensions of the carriers \(T(\xi_i)\) are bounded in the aggregate. But this follows from the fact that every thread \(\xi\) has the form \(\{f_i x\}\), \(x\in X\), while \(T(f_i x)\leq T(g_i x)\) and \(\dim T(g_i x)\leq k(x)\).

The proof of sufficiency can also be obtained from the theorem on coverings formulated above: in the inverse-limit compactum, as the sequence of coverings one may take the coverings composed of the inverse images of the principal stars of the complexes \(P_i\). However, it is easy to give a direct proof as well. For any natural number \(n\), consider the following sequence of closed subsets \(X_{i,n}\) of the polyhedra \(P_i\): let \(X_{1,n}\) be the \(n\)-dimensional skeleton of the polyhedron \(P_1\), and suppose the sets \(X_{i,n}\) for \(i\leq r\) have already been constructed; put
\[ X_{r+1,n}=(f_r^{r+1})^{-1}X_{r,n}\cap P_{r+1}^n, \]
where \(P_{r+1}^n\) denotes the \(n\)-dimensional skeleton of the polyhedron \(P_{r+1}\). The inverse limit \(X_n\) of the sequence \(\{X_{i,n}, f_i^j\}\) with the same mappings \(f_i^j\) (considered on \(X_{i,n}\)) is naturally embedded in the compactum \(X\). From the condition imposed on the threads it follows directly that
\[ X=\bigcup_n X_n. \]
The sets \(X_n\) are compact and \(\dim X_n\leq n\). Thus the theorem is completely proved.

Theorem 2. If a weakly countable-dimensional compactum \(X\) is represented as the inverse limit of a sequence of polyhedra \(\{Q_k, h_k^l\}\) satisfying Freudenthal’s conditions, then by passing to a subsequence, by an admissible deformation of the maps \(h_k^l\), and by subdividing the complexes \(Q_k\), one can pass from the original sequence to a sequence whose limit is still equal to \(X\), and whose nerves satisfy the condition of Theorem 1.

The proof of this theorem follows directly from the following proposition of Freudenthal: if a compactum \(X\) is represented as the inverse limit of two sequences of polyhedra \(\{P_i, f_i^j\}\) and \(\{Q_k, h_k^l\}\), then in them one can choose subsequences \(P_{i_n}\) and \(Q_{k_n}\) and construct maps
\[ h_{2n-1}: Q_{k_n}\to P_{i_n} \quad\text{and}\quad h_{2n}: P_{i_{n+1}}\to Q_{k_n} \]
in such a way that these maps will be simplicial with respect to certain subdivisions of the complexes \(P_{i_n}\) and \(Q_{k_n}\), the compositions \(h_{2n}h_{2n+1}\) and \(h_{2n-1}h_{2n}\) are obtained by an admissible deformation from the maps \(h_{k_n}^{k_{n+1}}\) and \(f_{i_n}^{i_{n+1}}\), and the limit of the alternating sequence of polyhedra
\[ P_{i_n}, Q_{k_n}, \qquad n=1,2,\ldots, \]
is equal to \(X\).

As \(\{P_i, f_i^j\}\) we take the sequence constructed in the preceding theorem, and apply Freudenthal’s proposition to the sequences \(\{P_i, f_i^j\}\) and \(\{Q_k, h_k^l\}\). From the resulting alternating sequence we choose the subsequence consisting of the polyhedra \(Q_{k_n}\), and take in them those same subdivisions with respect to which the maps \(h_{2n}\) are simplicial. This sequence will be the desired one.

In conclusion I express my sincere gratitude to Yu. M. Smirnov for valuable advice and comments.

Moscow State University
named after M. V. Lomonosov

Received
6 V 1960

REFERENCES CITED

¹ H. Freudenthal, Compositio Math., 4 (1937). ² Yu. M. Smirnov, Izv. AN SSSR, ser. matem., 23, No. 2 (1959). ³ E. Sklyarenko, DAN, 126, No. 6 (1959). ⁴ J. Nagata, Fund. Math., 48, No. 1 (1960). ⁵ P. S. Aleksandrov, Combinatorial Topology, Moscow–Leningrad, 1947.