MATHEMATICS

V. PONOMAREV

ON SOME APPLICATIONS OF PROJECTION SPECTRA TO THE THEORY OF TOPOLOGICAL SPACES

(Presented by Academician P. S. Aleksandrov on 26 I 1962)

In this paper a summary is given of certain results proved by direct application of the methods given in my papers \((^{1-3})\); these methods essentially belong to the theory of projection spectra. Some of these results are contained in the papers of Nagami \((^{6,7})\), published later and using essentially the same methods.

§ 1. In paper \((^2)\) the following order was considered in the set of all closed coverings, going back already to \((^4)\) and even to \((^5)\): let \(\alpha\) and \(\alpha'\) be two closed coverings; we say that \(\alpha'\) follows \(\alpha\) if \(\alpha'\) is inscribed in \(\alpha\) and each element of the covering \(\alpha\) is the sum of all elements of the covering \(\alpha'\) contained in it. A covering \(\alpha\) is called a partition or a canonical covering if it is locally finite and its elements are the closures of pairwise disjoint open sets of the space \(X\). In paper \((^2)\), and partly already in \((^1)\), the following notions were introduced.

A set \(\mathfrak A=\{\alpha\}\) of (arbitrary) coverings of any completely regular space is called: 1) refining, if for every point \(x\in X\) and every neighborhood \(Ox\) there is an \(\alpha_0\) such that the star of the point \(x\) in the covering \(\alpha_0\) is contained in \(Ox\); 2) strongly refining, if in the preceding definition one replaces the point \(x\) and its neighborhood \(Ox\) by an arbitrary closed \(F\subseteq X\) and its neighborhood \(OF\); 3) cofinally refining, if for every open covering \(\omega\) of the space \(X\) there is an \(\alpha\in\mathfrak A\) inscribed in it.

In \((^1)\) and further in \((^3)\) the following proposition was proved: under the order established in the set of closed coverings, the set of all finite canonical and, a fortiori, the set of all canonical coverings of a completely regular (respectively normal) space \(X\) will be a directed refining (respectively strongly refining) set.

To each directed set of canonical coverings \(\mathfrak A=\{\alpha\}\) of the space \(X\), with the natural mappings \(\omega_{\alpha}^{\alpha'}:\alpha'\to\alpha\) for \(\alpha'>\alpha\), there corresponds the spectrum \(S_{\mathfrak A}=\{|\alpha|,\omega_{\alpha}^{\alpha'}\}\) of the nerves \(|\alpha|\) of these coverings. The spectrum \(\dot S_{\mathfrak A}=\{\dot\alpha,\dot\omega_{\alpha}^{\alpha'}\}\), which we shall call the complete partition of the spectrum \(S_{\mathfrak A}\), consists of the complexes \(\dot\alpha\), where \(\dot\alpha\) is the zero-dimensional complex composed of all vertices of the complex \(|\alpha|\), with the same projections as in \(S_{\mathfrak A}\).

Suppose that in the space \(X\) a refining directed set of partitions \(\mathfrak A=\{\alpha\}\) is given. Then, by the method of paper \((^1)\), a space \(\dot X_0\) is constructed which is an everywhere dense subset of the limit space \(\dot X=\dot S_{\mathfrak A}\), and a perfect irreducible mapping \(f\) of the space \(\dot X_0\) onto the whole space \(X\). Let us recall how the space \(\dot X_0\) was constructed. A thread \(\dot x=\{e_\alpha\}\in\dot X\) was called marked if the sets \(A^\alpha\in\alpha\) corresponding to these vertices have a nonempty intersection. As a consequence of refinement

sequence \(\mathfrak A\), this intersection consists of a single point \(x\in X\). The totality of all marked threads \(\dot x=\{e_\alpha\}\) of the spectrum \(\widetilde S_{\mathfrak A}\) forms, by definition, the space \(\dot X_0\), and \(\operatorname{ind}\dot X_0=0\).

Let \(\dot x=\{e_\alpha\}\in \dot X_0\) be arbitrary; then the corresponding \(A^\alpha\in\alpha\) have an intersection consisting of a single point \(x\in X\). We obtain a mapping \(f\dot x=x:\dot X_0\to X\). As in (1), it is verified that this mapping is onto all of \(X\), and that \(f\) is a perfect* irreducible mapping of the space \(\dot X_0\) onto the (completely regular) \(X\).

It is easy to prove:

Lemma. If a \(T_1\)-space \(Y\) is the image of a \(T_1\)-space \(X\) under a closed continuous mapping \(f\), then the space \(Y\) is normal if and only if the space \(X\) is normal on every closed set** \(A\) of the form \(A=f^{-1}fA\).

Now it is proved (see (1)):

Theorem 1. Each of the following conditions is necessary and sufficient for the normality of the \(T_1\)-space \(X\):

A. The set of all (respectively, of all finite) partitions of the space \(X\) is directed and strongly refining.

B. The space \(X\) (of weight \(\tau\)) is the image under a perfect (irreducible) mapping \(f:X_0\to X\) of a completely regular and zero-dimensional, in the sense of \(\operatorname{ind}X_0=0\), space \(X_0\subseteq D^\tau\), which, for every set \(A\subseteq X\) of the form \(A=f^{-1}fA\), satisfies the condition \(\operatorname{Ind}_A X=0\).

The second part of this theorem, which strengthens the main result of the paper (1), is naturally supplemented by the proposition proved in (3).

Theorem 2. Each of the following conditions is necessary and sufficient in order that the \(T_1\)-space \(X\) be paracompact:

a) The set of all partitions of the space \(X\) is directed and confinally refining.

b) The space \(X\) is the image of a perfectly zero-dimensional space \(X_0\) under a perfect (irreducible) mapping*.

If the set of partitions \(\mathfrak A\) is confinally refining, then \(X\) is a paracompact space homeomorphic to the limiting space \(\widetilde S_{\mathfrak A}\) (see (3)); the perfectly zero-dimensional space \(\dot X_0\), of which the space \(\widetilde S\) is an irreducible perfect image, in this case coincides with the whole space \(\dot X=\widetilde S_{\mathfrak A}^{\dot{\ }}\).

Definition 1. A spectrum \(S_2=\{|\beta|,\omega_\beta^{\beta'}\}\) is called an amplification of the spectrum \(S_1=\{|\alpha|,\omega_\alpha^{\alpha'}\}\) if the following conditions are satisfied: 1) there exists a one-to-one similar mapping \(\alpha\to\beta_\alpha\) of the directed set of indices \(A=\{\alpha\}\) onto the directed set of indices \(B=\{\beta\}\) (here \(A\) may be regarded as a multiplication of \(B\)); 2) the set of all vertices \(\dot\beta_\alpha\) of the complex \(|\beta_\alpha|\) coincides with the set of all vertices \(\dot\alpha\) of the complex \(|\alpha|\), and \(|\beta_\alpha|\subseteq|\alpha|\).

* A mapping \(f:X\to Y\) is called perfect if it is continuous, closed, and bicompact in the sense that the inverse images \(f^{-1}y\) of all points \(y\in Y\) are bicompact.

* We call a space \(X\) normal on a closed set* \(A\subseteq X\) if, for every neighborhood \(OA\) of this set, there is a neighborhood \(O'A\) such that \([O'A]\subseteq OA\).

* A \(T_1\)-space is called perfectly zero-dimensional** if it is regular and, into each of its open covers, one can inscribe a cover consisting of pairwise nonintersecting open sets.

**** Every regular space \(X\) of weight \(\tau\) is the image of a completely regular space \(X_0\subseteq D^\tau\) under a continuous mapping \(f\). I do not know whether this mapping can be assumed perfect, or at least closed. The existence of the mapping \(f\) follows (with the aid of all the same methods) from the fact that in every regular space (and only in a regular space) the set of all finite partitions is directed and refining.

Definition 2. Two spectra \(S_1\) and \(S_2\) are called equivalent (or belonging to one and the same class) if their full refinements \(\dot S_1\) and \(\dot S_2\) are isomorphic.

Theorem 3. If the spectrum \(S_2\) is an enlargement of the spectrum \(S_1\), then there exists a single-valued irreducible perfect mapping*
\(\pi:\widetilde{\dot S}_1\to \widetilde{\dot S}_2^{*}\). In particular, there exists an irreducible perfect (“standard”) mapping \(\pi_X\) of the limiting space \(\widetilde{\dot S}=\dot X\) (where \(\dot S\) is the full refinement of the spectrum \(S\)) onto the space \(\widetilde S=X\) of the spectrum \(S=\{\alpha,\omega_\alpha^a\}\). If two spectra \(S_1\) and \(S_2\) are equivalent, then and only then their spaces \(X_1\) and \(X_2\) are mapped onto each other multivalently, perfectly and irreducibly (in the sense of paper \((^3)\)) by the formula

\[ f=\pi_{X_2} f^{\cdot}(\pi_{X_1})^{-1}, \]

where \(f^{\cdot}\) is some homeomorphism of the space \(\widetilde{\dot S}_1\equiv \widetilde{\dot S}_2\equiv \dot X\) onto itself, and \(\pi_{X_2},\pi_{X_1}\) are the corresponding standard mappings of \(\dot X\) onto \(X_2\) and \(X_1\); these mappings may be interpreted as a passage from the spectrum \(\dot S_i\) to the spectrum \(S_i,\ i=1,2\), which is an enlargement of the spectrum \(\dot S_i\).

§ 2. Definition 1. Let in \(X\) (an arbitrary normal space) there be a directed set \(\mathfrak A\) of decompositions of multiplicity \(\leq n+1\): a) refining; b) strongly refining; c) cofinally refining (in the last case the space \(X\) is paracompact). Then we say that the approximation dimension \(dX\) (in case a)), the large approximation dimension \(DX\) (in case b)), the cofinal approximation dimension \(\Delta X\) (in case c)) does not exceed \(n\). Naturally, \(dX\) itself, respectively \(DX\) and \(\Delta X\), is the least \(n\geq 0\) (if it exists) satisfying this condition. If there is no such \(n\), then the corresponding dimension is taken to be equal to \(\infty\).

Entirely by the methods of paper \((^2)\) one proves

Theorem 4. In any normal space the following relations between dimensions hold:

\[ \operatorname{ind}X\leq dX,\qquad \operatorname{ind}X\leq \operatorname{Ind}X\leq DX,\qquad \dim X\leq \Delta X. \]

If the dimension \(\dim X=\Delta X\), then

\[ \operatorname{ind}X\leq \operatorname{Ind}X\leq \dim X=\Delta X. \]

Definition 2 (basic) \((^2)\). A space \(X\) is called perfectly \(n\)-dimensional if \(\dim X=\Delta X=n\).

Corollary of Theorem 4. If a strongly paracompact space \(X\) is perfectly \(n\)-dimensional, then

\[ \operatorname{ind}X=\operatorname{Ind}X=\dim X=\Delta X=n. \]

Theorems 2 and 3 of paper \((^2)\) carry over to arbitrary normal spaces with the aid of the constructions given above (taken from papers \((^1,^2)\)). We obtain the propositions:

Theorem 5′. A completely regular space \(X\) (of weight \(\tau\)) has \(dX\leq n\) if and only if there exists \(X_0\subseteq D^\tau\) (and hence \(\operatorname{ind}X_0=0\)) and an \((n+1)\)-fold irreducible perfect mapping \(f\) of the space \(X_0\) onto \(X\).

Theorem 5″. \(\Delta X\leq n\) if and only if \(X\) is a perfect \((n+1)\)-fold image of some perfectly zero-dimensional space** \(X_0\).

* In this case the mapping \(\pi\) may be interpreted as a passage from the spectrum \(S_1\) to the spectrum \(S_2\) in the sense of paper \((^3)\).

** See papers \((^1,^3)\).

We note that from the results of Morita \(^6\) and Theorem 6 of the present paper there follows

Theorem 6. A metric space \(X\): \(\dim X = n\) is perfectly \(n\)-dimensional.

Moscow State University
named after M. V. Lomonosov

Received
22 XII 1961

REFERENCES

\(^1\) V. Ponomarev, DAN, 132, 1269 (1960).
\(^2\) P. Aleksandrov, V. Ponomarev, Siberian Math. J., 1, No. 1, 3 (1960).
\(^3\) V. Ponomarev, DAN, 143, No. 4 (1962).
\(^4\) I. V. Proskuryakov, Uch. zap. Mosk. univ., 148, Mathematics, 1, 219 (1951).
\(^5\) P. Aleksandrov, Combinatorial Topology, Ch. 6, Moscow—Leningrad, 1947.
\(^6\) K. Nagami, Proc. Japan Acad., 37, No. 4, 189 (1961).
\(^7\) K. Nagami, Proc. Japan Acad., 37, No. 4, 193 (1961).