UDC 513.83

MATHEMATICS

V. ZAĬTSEV

ON PROJECTION SPECTRA AND BICOMPACT EXTENSIONS

(Presented by Academician P. S. Aleksandrov on 25 VIII 1966)

The notion of a projection spectrum as the inverse spectrum of a countable set of finite complete simplicial complexes with simplicial mappings as projections was introduced by P. S. Aleksandrov (¹); this notion was freed by A. G. Kurosh (²) from the requirement, entering into it, that the set of complexes be countable; finally, the notion of a projection spectrum received full generality in the works of V. I. Ponomarev (³, ⁴) (an arbitrary directed set of simplicial complexes of arbitrary cardinality). The Aleksandrov–Kurosh theorem states:

Every bicompactum (bicompact Hausdorff space) is the limit of some projection spectrum; conversely, the limit of any finite (i.e. consisting of finite complexes) spectrum is a bicompact \(T_1\)-space, possibly non-Hausdorff.

V. I. Ponomarev proved that every paracompactum (paracompact Hausdorff space) is the limit of some spectrum (but no longer a finite one). In fact these authors did more: they showed that every space considered by them (a bicompactum in Kurosh, a paracompactum in Ponomarev) is the limit of a certain quite definite spectrum, namely the maximal (respectively, maximal finite) spectrum of the given space. Under this name V. I. Ponomarev was the first to define with complete clarity the spectrum whose complexes are the nerves of all, respectively all finite, coverings of the given space. Here by a covering is meant any locally finite covering \(\alpha\), whose elements are canonical closed sets \(A_\lambda^\alpha = [U_\lambda^\alpha]\) with disjoint open kernels \(U_\lambda^\alpha = I A_\lambda^\alpha\). The order and the projections are defined naturally: \(\alpha' > \alpha\), if every element \(A_\lambda^{\alpha'}\) of the covering \(\alpha'\) is contained in some (obviously unique) element \(A_\lambda^\alpha\) of the covering \(\alpha\); by associating with the element \(A_\lambda^{\alpha'}\) the element \(A_\lambda^\alpha\) of the covering \(\alpha\) that contains it, we obtain a simplicial mapping (projection) of the nerve \(\alpha'\) onto the nerve \(\alpha\). For bicompact spaces the notions of maximal spectrum and maximal finite spectrum coincide, since a locally finite covering of a bicompact space is always finite. Thus every bicompactum (Kurosh) and every paracompactum (Ponomarev) is the limit of its maximal projection spectrum. On the other hand, the limit of every projection spectrum is a \(T_1\)-space, bicompact in the case of finite spectra; however, not every bicompact \(T_1\)-space is the limit of its maximal spectrum.

In this paper, by topological spaces we shall always mean \(T_1\)-spaces. Recall that a \(T_1\)-space is called semiregular if the canonical open sets form a base in it (or, equivalently, the canonical closed sets form a closed base). For brevity, we shall call semiregular \(T_1\)-spaces \(T_\xi\)-spaces. \(T_\xi\)-spaces need not be Hausdorff; at the same time not every Hausdorff space is a \(T_\xi\)-space.

Finally, the following two results substantially complete the whole picture:

The limit of the maximal finite spectrum of a normal space \(X\) is the Čech extension \(\beta X\) (P. S. Aleksandrov).

The limit of the maximal finite spectrum of any regular space \(X\) is the bicompact \(T_1\)-extension \(\omega_\chi X\) of Wallman type introduced by Ponomarev \((^5)\): the points of the space \(\omega_\chi X\) are \(\chi\)-systems, i.e. maximal centered systems of canonical closed sets of the space \(X\); the topology in \(\omega_\chi X\) is the usual Wallman topology. We shall return to this last result below.

In the present note the following facts are established.

Theorem 1. If the maximal finite spectrum \(S_X\) of a given \(T_1\)-space \(X\) exists (i.e. if the finite decompositions of the space \(X\), with their natural order, form a directed set), then the limiting space \(\widetilde S_X\) of the spectrum \(S_X\) is naturally homeomorphic to the space \(\omega_\chi X\).

The proof of this theorem is based on the following considerations. Along with the Wallman—Ponomarev space we define the space \(\omega_\pi X\), whose points are \(\pi\)-systems, i.e. maximal centered systems of sets \(P\), each of which is the intersection of a finite number of canonical closed sets (the topology is the same).

First of all one proves

Lemma 1. The spaces \(\omega_\pi X\) and \(\omega_\chi X\) are naturally homeomorphic to each other.

For the proof of Lemma 1 we take an arbitrary \(\pi\)-system \(\xi\) and, in it, the subsystem \(\chi\xi\) consisting of canonical closed sets alone.

It turns out that \(\xi\) consists of all finite intersections of elements of the system \(\chi\xi\), and that \(\chi\xi\) is a \(\chi\)-system.

Assigning to each \(\pi\)-system the unique \(\chi\)-system contained in it, we obtain a natural homeomorphism between the spaces \(\omega_\pi X\) and \(\omega_\chi X\).

V. I. Ponomarev, from analogous considerations, even derives a homeomorphism between his space \(\omega_\chi X\) and the original Wallman space \(\omega X\). However, his reasoning on this point does not seem convincing to me: the spaces \(\omega X\) and \(\omega_\chi X\), generally speaking, are distinct.

For the proof of Theorem 1 one can, relying on Lemma 1, prove that the space \(\widetilde S_X\) is homeomorphic to the space \(\omega_\pi X\), and apply Ponomarev’s method to this in the appropriate way.

Let \(\xi=\{P_\lambda\}\) be a point of the space \(\omega_\pi X\). In each decomposition \(\alpha=\{A_1^\alpha,\ldots,A_{s_\alpha}^\alpha\}\) of the space \(X\) there is an element \(A_i^\alpha\) which intersects all \(P_\lambda\in\xi\), and then \(A_i^\alpha\in\xi\). Let \(A_{i_0}^\alpha,\ldots,A_{i_r}^\alpha\) be all the elements of the covering \(\alpha\) each of which intersects all \(P_\lambda\in\xi\). Then \(A_{i_0}^\alpha,\ldots,A_{i_r}^\alpha\) are contained in \(\xi\), and, consequently,

\[ A_{i_0}^\alpha\cap\cdots\cap A_{i_r}^\alpha\ne \Lambda \]

—the elements \(A_{i_0}^\alpha,\ldots,A_{i_r}^\alpha\) form a vertex of the complex \(t_\alpha=t_\alpha(\xi)\) of the nerve of \(\alpha\). Performing this construction for each \(\alpha\), we obtain a thread \(\eta(\xi)=\{t_\alpha(\xi)\}\) of the spectrum \(S_X\), and this thread proves to be maximal. Thus, to each point \(\xi\in\omega_\pi X\) there corresponds a point \(\eta=\eta(\xi)\in\widetilde S_X\), and the resulting mapping of the space \(\omega_\pi X\) into the space \(\widetilde S_X\) is, as can be shown, the required homeomorphism between \(\omega_\pi X\) and \(\widetilde S_X\).

Since for every \(T_\xi\)-space \(X\) the maximal finite spectrum exists, Theorem 1 implies

Corollary. The limit of the maximal finite spectrum of every \(T_\xi\)-space \(X\) is the space \(\omega_\chi X\).

We now pass to the question: when is a given bicompact space \(X\) the limit of its maximal finite spectrum?

It is proved first of all:

Theorem 2a. If \(X\) is a bicompact \(T_{\zeta}\)-space, then the natural mapping

\[ \varphi:\ X \to \check S_X \]

is a homeomorphism.

Here by the natural mapping \(\varphi:\ X \to \check S_X\) is meant the mapping constructed as follows: for each point \(x \in X\), in each partition

\[ \alpha=\{A_1^\alpha,\ldots,A_{s_\alpha}^\alpha\} \]

we take the set of all elements of this partition containing the point \(x\). We obtain a simplex \(t_\alpha(x)\) of the nerve \(|\alpha|\). The simplexes \(t_\alpha(x)\), constructed for all \(\alpha\), form a maximal thread of the spectrum \(S_X\) of the space \(X\), i.e. a point of the space \(\check S_X\), which is thus put into correspondence with the point \(x\). Next, we have

Theorem 2b. If the space \(X\) is naturally homeomorphic to the limit of its maximal finite spectrum \(\check S_X\), then it is a \(T_{\zeta}\)-space.

Combining Theorems 2a and 2b, we obtain the following result:

Theorem 2. A bicompact space \(X\) is naturally homeomorphic to the limit of its maximal spectrum \(\check S_X\) if and only if \(X\) is a \(T_{\zeta}\)-space.

Obviously, every regular space is a \(T_{\zeta}\)-space in which the canonical closed sets form a net.*

Let us call a \(T_\lambda\)-space any \(T_{\zeta}\)-space in which the canonical closed sets form a pseudonet.*

For what follows we shall need

Lemma 2. Let \(X\) be a \(T_\lambda\)-space, \(\alpha=\{A_1^\alpha,\ldots,A_{s_\alpha}^\alpha\}\) its finite partition. For each closed canonical set \(A\) in \(X\), denote by \(\bar A\) the closure of \(A\) in \(\omega_\varkappa X\).

Then, if \(\bar A_{i_1}^\alpha \cap \cdots \cap \bar A_{i_r}^\alpha \ne \Lambda\), then also \(A_{i_1}^\alpha \cap \cdots \cap A_{i_r}^\alpha \ne \Lambda\).

Indeed, if \(\xi=\{A_\lambda\}\in \bar A_{i_1}^\alpha \cap \cdots \cap \bar A_{i_r}^\alpha\), then \(A_k^\alpha \in \xi\) for \(k=i_1,\ldots,i_r\); hence
\[ A_{i_1}^\alpha \cap \cdots \cap A_{i_r}^\alpha \ne \Lambda. \]

From this simple lemma it follows that the nerves of the coverings \(\alpha=\{A_1^\alpha,\ldots,\ldots,A_{s_\alpha}^\alpha\}\) and \(\bar\alpha=\{\bar A_1^\alpha,\ldots,\bar A_{s_\alpha}^\alpha\}\), respectively, of the spaces \(X\) and \(\omega_\varkappa X\), are isomorphic, and then the maximal finite spectra of the spaces \(X\) and \(\omega_\varkappa X\) are also isomorphic. Hence and from Theorem 1 we obtain

Theorem 3. For any \(T_\lambda\)-space \(X\), the space \(\omega_\varkappa X\) is homeomorphic to the limit of its maximal finite spectrum. It turns out that this homeomorphism is natural. Therefore, relying on Theorem 2, we conclude that \(\omega_\varkappa X\) is a \(T_{\zeta}\)-space.

It can also be shown that \(\omega_\varkappa X\) is an extension of the \(T_\lambda\)-space \(X\). Thus, we have:

Theorem 4. For any \(T_\lambda\)-space \(X\) (in particular, for any regular space), the space \(\omega_\varkappa X\) is a bicompact \(T_{\zeta}\)-extension.

It follows from Theorem 4 that every \(T_\lambda\)-space has a bicompact—

* A net of a space \(X\) in the sense of A. V. Arhangel’skii is any system \(\mathfrak M\) of sets \(M \subseteq X\) satisfying the condition: whatever the point \(x\) and its neighborhood \(Ox\), there exists an \(M \in \mathfrak M\) such that \(x \in M \subseteq Ox\). A system \(\mathfrak M\) of sets \(M \subseteq X\) is called a pseudonet if, by adjoining to it all possible finite intersections \(M_1 \cap \cdots \cap M_s\) of its elements, we obtain a net.

a $T_\xi$-extension. The converse assertion is also true. In other words:

Theorem 5. In order that a space $X$ have a bicompact $T_\xi$-extension, it is necessary and sufficient that $X$ be a $T_\lambda$-space.

The necessity of the condition contained in this theorem follows from the following two lemmas.

Lemma 3. The canonical closed sets in a bicompact $T_\xi$-space form a pseudonet.

Lemma 4. If the canonical closed sets form a pseudonet in a bicompact extension $\overline X$ of a space $X$, then the canonical closed sets of the space $X$ also form a pseudonet.

Consider some $T_\lambda$-space $X$ and its bicompact $T_\xi$-extension $\xi X$. In view of the fact that the closure operator in the space $\xi X$ establishes a one-to-one correspondence between the canonical closed sets of the spaces $X$ and $\xi X$, and also establishes a one-to-one correspondence between the finite partitions $\alpha$ and $\overline{\alpha}$ of these spaces, the nerve of the partition $\overline{\alpha}$ is an “extension” of the nerve $\alpha$ (the two nerves have one and the same set of vertices, but $\overline{\alpha}$ may have additional simplexes in comparison with $\alpha$). Hence the spectrum of the space $\xi X$ is also an extension of the spectrum of the space $X$ and of the spectrum of the space $\omega_\lambda X$ isomorphic to it.

Each point of the space $\omega_\lambda X$ may be regarded as a maximal thread $\xi^x$ of the spectrum $S_X = S_{\omega_\lambda X}$; this thread is a thread $\xi^x$ also of the spectrum $S_{\xi X}$ of the space $\xi X$, generally speaking, not maximal. In the case when the thread is not a maximal thread, we take the unique maximal thread containing it. This establishes a natural mapping $\xi: \omega_\lambda X \to \xi X$ of the space $\omega_\lambda X$ into $\xi X$, which we call the spectral mapping. Under this mapping every $\overline A_i^\alpha = [A_i^\alpha]_{\omega_\lambda X}$ is mapped into $[A_i^\alpha]_{\xi X} = \xi \overline A_i^\alpha$.

It follows easily from this that the spectral mapping

\[ \xi:\ \omega_\lambda X \to \xi X \]

is $\theta$-continuous. Further, it is a mapping of the space $\omega_\lambda X$ onto the whole space $\xi X$; finally, all points of $X$ remain fixed under the mapping $\xi$.

Thus we have:

Theorem 6. Among all bicompact $T_\xi$-extensions of a given regular space $X$, the Wallman–Ponomarev extension $\omega_\lambda X$ is the unique maximal one in the sense that the spectrum of the space $\xi X$ is an extension of the spectrum of the space $\omega_\lambda X$; there exists a natural mapping of the space $\omega_\lambda X$ onto $\xi X$. This mapping is $\theta$-continuous and leaves all points of $X$ fixed.

The present work was carried out under the supervision of P. S. Aleksandrov, to whom I express my sincere gratitude.

Moscow State University
named after M. V. Lomonosov

Received
17 VIII 1966

REFERENCES

1. P. S. Aleksandrov, Ann. Math., 30, 101 (1928).
2. A. G. Kurosh, Compositio math., 2, 471 (1935).
3. V. I. Ponomarev, DAN, 143, No. 1, 46 (1962).
4. V. I. Ponomarev, Matem. sborn., 60, 1, 89 (1963).
5. V. I. Ponomarev, Sibirsk. matem. zhurn., 5, 6, 1333 (1964).