Mathematics

B. Pasynkov

On a Class of Transitive One-to-One Spectra for Bicompacta

(Presented by Academician P. S. Aleksandrov, 19 XI 1959)

In this note we solve the problem posed by P. S. Aleksandrov in 1935 of establishing a one-to-one correspondence between spectra of a certain class and bicompacta.

Definition 1. By a spectrum (S={X_\alpha,\omega_\alpha^\beta}) we shall mean a set of complexes (X_\alpha) (the totality of indices (\alpha) forms a directed partially ordered set) and simplicial mappings
(\omega_\alpha^\beta:X_\beta\to X_\alpha), defined whenever (\beta>\alpha). The mappings (\beta>\alpha), called projections, satisfy the condition of transitivity:
[
\omega_\alpha^\beta\omega_\beta^\gamma=\omega_\alpha^\gamma
\quad\text{for } \gamma>\beta>\alpha .
]

Remark. The complexes (X_\alpha) are not assumed to be complete, i.e., the set (X_\alpha) need not contain, together with a simplex (t_\alpha), all its faces. If (t_\alpha) is a face of (t'\alpha), then we shall write (t\alpha\ll t'\alpha). In this way (X\alpha) is turned into a partially ordered set, which may also be understood as a (T_0)-space ((^1)).

Definition 2. A set (t={t_\alpha}), one simplex (t_\alpha) from each (X_\alpha), will be called a projection set if, for (\beta>\alpha), we always have
[
t_\alpha=\omega_\alpha^\beta t_\beta .
]
A projection set (t={t_\alpha}) will be called a thread if there is no projection set (t'={t'\alpha}), distinct from (t), such that for every (\alpha), (t\alpha\ge t'\alpha). In what follows, the complexes (X\alpha) are everywhere assumed to be finite, and the projections (\omega_\alpha^\beta) are mappings “onto.”

Definition 3. A simplex (t_\alpha\in X_\alpha) will be called minimal if (X_\alpha) contains no proper faces of (t_\alpha).

Definition 4. We shall call a simplex (t_\alpha) essential if in some (X_\beta) ((\beta\ge\alpha)) there is a minimal simplex (t_\beta) projecting onto (t_\alpha)
[
(\omega_\alpha^\beta t_\beta=t_\alpha).
]

Theorem 1. The condition that the simplex (t_\alpha) be essential is necessary and sufficient in order that “a thread pass through it” (i.e., that there exist a thread containing among its elements the simplex (t_\alpha)).

Definition 5. A spectrum (S) will be called separated from above if, for any two essential simplices (t_\alpha) and (t'\alpha), separated from below in the sense of the equality
[
[t\alpha]\cap [t'\alpha]=\Lambda,
]
there exists (\beta>\alpha) for which “separation from above” takes place in the sense
[
O\beta(\omega_\alpha^\beta)^{-1}[t_\alpha]\cap
O_\beta(\omega_\alpha^\beta)^{-1}[t'\alpha]=\Lambda,
]
where, as always, (O\beta) denotes the star in the complex (X_\beta), and square brackets denote closure (in the present case in the complex (X_\alpha)).

Theorem 2. The condition that the spectrum (S) be separated from above is equivalent to the Hausdorff condition for this spectrum, namely, that for any two threads (t={t_\alpha}) and (t'={t'\alpha}) there exists an (\alpha) such that (t\alpha) and (t'\alpha) have disjoint stars in (X\alpha).

By definition, the threads are the points of the limit space (X) of the spectrum (S).

The topology in (X) is introduced as follows: a neighborhood (U_{\alpha_0}t) of the thread (t={t_\alpha}), by definition, consists of all threads (t'={t'\alpha}) for which (t').}\in Ot_{\alpha_0

If to each thread (t={t_\alpha}) we put in correspondence (t_\alpha) (its (\alpha)-th coordinate), then we obtain a continuous mapping (projection) (\wp_\alpha) of the space (X) into (X_\alpha). It is clear that if the spectrum (S) is essential, i.e. all simplexes of its complexes are essential, then (\wp_\alpha) and (\wp_\alpha^\beta) are mappings “onto.” For transitive spectra (\wp_\alpha=\wp_\alpha^\beta\wp_\beta) for (\beta>\alpha).

In what follows, by a spectrum is meant a Hausdorff essential spectrum.

Definition 6. We shall call a subspectrum (S'={X_{\alpha'},\wp_{\alpha'}^{\beta'}}) of the spectrum (S={X_\alpha,\wp_\alpha^\beta}) a nonseparating subspectrum of the spectrum (S), if: 1) in (S) there is an (X_{\alpha_0}), and in it such simplexes (t_{1\alpha_0}) and (t_{2\alpha_0}), that ([t_{1\alpha_0}]\cap [t_{2\alpha_0}]=\Lambda); 2) for every (\beta>\alpha_0) in (X_\beta) there exist (t_{1\beta}) and (t_{2\beta}), for which (\wp_{\alpha_0}^{\beta}t_{1\beta}=t_{1\alpha_0}), (\wp_{\alpha_0}^{\beta}t_{2\beta}=t_{2\alpha_0}), and which are “glued together” in (S'), i.e. (\wp_{\alpha'}^\beta t_{1\beta}\leq \wp_{\alpha'}^\beta t_{2\beta}) for (X_{\alpha'}\in S'). If these conditions are not fulfilled, then the subspectrum is naturally called separating.

Definition 6 can also be given the following form: a subspectrum (S') of a spectrum (S) is separating if and only if for any two threads (t_1) and (t_2) of the spectrum (S) there exists in (S') such a complex (X_{\alpha'}) that the simplexes (t_{1\alpha'}=t_1\cap X_{\alpha'}) and (t_{2\alpha'}=t_2\cap X_{\alpha'}) have disjoint stars in (X_{\alpha'}).

Theorem 3. In order that a subspectrum (S') of a spectrum (S) give the same limit space as (S), it is necessary and sufficient that (S') be a separating subspectrum.

Definition 7. We shall say that a spectrum (\widetilde S) is obtained from a spectrum (S={X_\alpha,\wp_\alpha^\beta}) by a weakening of the order, if (\widetilde S) consists of the same complexes (X_\alpha) as (S), from (\beta>\alpha) in (\widetilde S) there follows (\beta>\alpha) in (S), and the projections in both spectra are the same.

Definition 8. Given a spectrum (S={X_\alpha,\wp_\alpha^\beta}). We shall call complexes (X_\alpha) and (X_\beta) isomorphic with respect to the spectrum (S) ((S)-isomorphic), if between (X_\alpha) and (X_\beta) one can establish an isomorphism such that, under (t_\alpha\leftrightarrow t_\beta), the conditions (\wp_\alpha^\gamma t_\gamma=t_\alpha) and (\wp_\beta^\gamma t_\gamma=t_\beta) are equivalent for every (\gamma>\beta,\alpha).

Basic definition. A spectrum (\Sigma) is called extremal if it satisfies the following conditions:

1) The spectrum (\Sigma) contains no two (\Sigma)-isomorphic complexes (minimality condition).

2) The spectrum (\Sigma) cannot be obtained from another spectrum by weakening the order (first maximality condition).

3) The spectrum (\Sigma) is not a separating subspectrum of any spectrum satisfying conditions 1) and 2) (second maximality condition).

Main theorem. Every bicompactum (X) is the limit space of some—and, up to isomorphism, of only one—extremal spectrum.

Remark. It is known ((^1)) that every Hausdorff transitive spectrum of finite complexes, in particular an extremal one, has as its limit space a bicompactum. Therefore our main theorem establishes a one-to-one correspondence between bicompacta and their extremal spectra, i.e. solves the problem posed at the beginning of the paper.

The author expresses sincere gratitude to P. S. Aleksandrov, under whose supervision the work was carried out.

Received
18 XI 1959

CITED LITERATURE

1. P. S. Aleksandrov, Uspekhi Mat. Nauk, 2, no. 1 (1947).