MATHEMATICS

V. PONOMAREV

ON PARACOMPACT SPACES AND THEIR CONTINUOUS MAPPINGS

(Presented by Academician P. S. Aleksandrov, 23 XI 1961)

§ 1. In this note it is first of all established (Theorem 1 and the basic definition in § 2) that all paracompacts, and only they, admit a sufficiently good approximation by projection spectra (in the classical sense established by P. S. Aleksandrov \((^{1a})\) and generalized by A. G. Kurosh \((^{1b})\)). From this theorem it is then deduced (Theorem 2) that all paracompacts, and only they, are images of perfectly zero-dimensional spaces under perfect continuous irreducible** mappings. Moreover, for each paracompact \(X\) the perfectly zero-dimensional space \(\dot X\), whose image is the paracompact \(X\), is determined by a certain natural one-to-one construction; therefore I call this space \(\dot X\) an absolute preimage, or simply the absolute, of the space \(X\). It turns out (Theorem 5) that two paracompacts can be perfectly and irreducibly (but, generally speaking, many-valuedly) mapped onto one another if and only if they have one and the same absolute; moreover, every one-to-one irreducible perfect mapping \(f: X \to Y\) determines a certain homeomorphism \(\dot f: \dot X \to \dot Y\) between the absolutes \(\dot X\) and \(\dot Y\), and in turn is determined by it (Theorem 4).

§ 2. A projection spectrum is a directed set
\(S=\{\alpha,\mathfrak d_{\alpha}^{\alpha'}\}\) of simplicial complexes \(\alpha\) with simplicial mappings (projections) \(\mathfrak d_{\alpha}^{\alpha'}\) of the complex \(\alpha'\) onto \(\alpha\) (for all \(\alpha'>\alpha\)). In addition the transitivity condition is fulfilled: if \(\alpha''>\alpha'>\alpha\), then
\(\mathfrak d_{\alpha}^{\alpha''}=\mathfrak d_{\alpha}^{\alpha'}\mathfrak d_{\alpha'}^{\alpha''}\). A thread of the spectrum is a system \(\xi=\{t_\alpha\}\) of simplices, one \(t_\alpha\) from each \(\alpha\), having the property that for \(\alpha'>\alpha\) one always has \(\mathfrak d_{\alpha}^{\alpha'}t_{\alpha'}=t_\alpha\). A thread \(\xi'=\{t'_\alpha\}\) embraces the thread \(\xi=\{t_\alpha\}\) if for every \(\alpha\) we have \(t_\alpha \geq t_\alpha'\) (this means that \(t_\alpha\) is a proper or improper face of the simplex \(t_\alpha'\)). A thread \(\xi\) is called maximal if it has no thread embracing it that is distinct from it. The space \(\widetilde S=\lim S\) of the spectrum \(S\) is the set of all maximal threads. The topology on this set is introduced by means of a base consisting of elementary open sets \(Ot_\alpha\); here \(Ot_\alpha\)

* A space \(\dot X\) is called perfectly zero-dimensional if into every open covering \(\omega\) of the space \(\dot X\) one can inscribe a covering \(\omega'\) whose elements are pairwise disjoint (open-and-closed) sets.

** A one-to-one mapping \(f: X\to Y\) is called perfect if it is closed, continuous, and the preimages \(f^{-1}y\) of all points \(y\in Y\) are bicompact. A many-valued mapping is perfect \((^2)\) if, in addition, the images \(fx\) of all points \(x\in X\) are bicompact. For a many-valued perfect mapping \(f: X\to Y\) there exists a space \(Z\) and one-to-one mappings \(f_X: Z\to X\) and \(f_Y: Z\to Y\) such that \(f=f_Y f_X^{-1}\).

*** A one-to-one mapping \(f\) of a space \(X\) onto \(Y\) is called irreducible if, whatever closed subset \(A\subset X,\ A\ne X\), we always have \(fA\ne Y\). A many-valued perfect mapping \(f: X\to Y\) is called irreducible if a space \(Z\) and irreducible one-to-one perfect mappings \(f_X: Z\to X,\ f_Y: Z\to Y\) can be found in such a way that \(f=f_Y f_X^{-1}\).

is the set of all maximal threads \(\xi'\) for which, for a given fixed \(\alpha\), we have \(t_\alpha \geq t'_\alpha,\ t'_\alpha \in \xi'\). The space \(\widetilde S\) with this topology is always a \(T_1\)-space. Let \(e_\alpha\) be an arbitrary vertex of the given fixed complex \(\alpha \in S\). By \(\Phi_{e_\alpha}\) we denote the set of all maximal threads \(\xi\) for which (for the given \(\alpha\)) we have \(e_\alpha \leq t_\alpha \in \xi\). The sets \(\Phi_{e_\alpha}\) are closed. For a given fixed \(\alpha\), the sets \(\Phi_{e_\alpha}\), where \(e_\alpha\) runs through the totality of all vertices of the complex \(\alpha\), form a closed covering \(\varphi_\alpha\) of the space \(\widetilde S\).

Remark. The nerve of the covering \(\varphi_\alpha\) is always a subcomplex of the complex \(\alpha\). If the spectrum \(S\) is complete*, then the nerve of the covering \(\varphi_\alpha\) coincides with the complex \(\alpha\).

Definition. A spectrum \(S\) is called regular if, whatever the point \(\xi=\{t_\alpha\}\) and the given \(\alpha\) may be, there exists an \(\alpha'>\alpha\) such that, for \(t_{\alpha'}\in\xi\), \(t_{\alpha'}=|e^0_{\alpha'},\ldots,e^r_{\alpha'}|\), we have

\[ \bigcup_{i=1}^r \Phi_{e^i_{\alpha'}} \subseteq O t_\alpha . \]

The space of a regular spectrum is always a regular space.

Basic definition. A spectrum \(S\) is called uniform if, whatever the covering \(\omega=\{U\}\) of its space \(S\) by elementary open sets may be, there exists an \(\alpha\in S\) such that the covering \(\varphi_\alpha\) is inscribed in the covering \(\omega\).

Theorem 1. The space of every regular uniform spectrum is paracompact; conversely, every paracompact space is the space of a certain complete regular uniform spectrum.

The proof of the first assertion rests on the fact that every covering \(\varphi_\alpha\) is locally finite.

For the proof of the second assertion, in the paracompact \(X\) one takes the directed (in the natural way) system of all decompositions. Then the nerves of these decompositions, with the natural projections, form the required spectrum \(S\). This spectrum \(S\) is called the spectrum of the paracompact** \(X\). It is proved that the spectrum \(S\) is uniform, regular, and complete, and that its space is homeomorphic to the paracompact \(X\).

§ 3. Let \(S\) be the spectrum of a paracompact \(X\).

Proposition A. Each thread \(\xi'=\{t'_\alpha\}\) of the spectrum \(S\) is contained in a unique maximal thread \(\xi=\{t_\alpha\}\).

Proposition B. Each thread \(\xi=\{t_\alpha\}\) of the spectrum \(S\) envelops some thread \(\xi_0=\{e_\alpha\}\), where \(e_\alpha\) is a vertex of the simplex \(t_\alpha\).

Further, for any \(\alpha\in S\), by \(\dot\alpha\) we denote the zero-dimensional complex consisting of all vertices of the complex \(\alpha\). Keeping the projections \(\mathfrak D^{\alpha'}_{\alpha}\) of the spectrum \(S\) (but considering them as mappings of the complex \(\dot\alpha'\) onto \(\dot\alpha\)), we obtain a spectrum \(\dot S=\{\dot\alpha,\mathfrak D^{\alpha'}_{\alpha}\}\), uniquely determined by the spectrum \(S\) (and, consequently, by the paracompact \(X\)), called the complete relaxation of the spectrum \(S\). Its space (the uniquely determined paracompact \(X\)) we denote by \(\dot X\): \(\dot X=\lim \dot S\), and call it the absolute of the space \(X\). The elementary open and elementary closed sets of the spectrum \(\dot S\) coincide—these are the sets (open-and-closed)

\[ \widetilde e_{\alpha_0}=E(\xi'=\{\dot e_\alpha\},\ \dot e_{\alpha_0}=e_{\alpha_0}). \]

Each thread \(\dot x=\{e_\alpha\}\) of the spectrum \(\dot S\) is maximal; at the same time it is a thread of the spectrum \(S\).

* The spectrum \(S=\{\alpha,\mathfrak D^{\alpha'}_{\alpha}\}\) is called complete if, for every simplex \(t_\alpha\in\alpha\), there exists a maximal thread \(\xi\ni t'_\alpha\geq t_\alpha\).

** A decomposition is a locally finite covering whose elements are the closures of pairwise intersecting open sets.

Assigning to each thread $\dot{x}=\{e_\alpha\}\in \dot{X}$ the unique (by Proposition A) maximal thread $x$ of the spectrum $S$ that contains it, we obtain a mapping $\pi_X:\dot{X}\to X$, which is (by Proposition B) a mapping of the space $\dot{X}$ onto the space $X$.

Theorem 2′. The mapping $\pi_X$ just constructed (cf. (3)) of the absolute $\dot{X}$ of the space $X$ onto the space $X$ is a perfect irreducible mapping.

Theorem 2″. The absolute $\dot{X}$ of every paracompactum $X$ is a perfectly zero-dimensional (and hence strongly paracompact) space, for which
\[ \dim \dot{X}=\operatorname{ind}\dot{X}=\operatorname{Ind}\dot{X}=0. \]

Since, on the other hand, the image of every paracompactum under a perfect mapping is always a paracompactum, the following characterization of paracompact spaces follows from Theorems 2′ and 2″:

Theorem 2. Among regular spaces, the paracompacta, and only they, are images of perfectly zero-dimensional spaces under perfect irreducible mappings.

§ 4. In this and the following paragraphs it will be convenient for us to denote by $\alpha$ an arbitrary covering of the paracompactum $X$, by $|\alpha|$ its nerve, and by $\dot{\alpha}$ the set of all vertices of the nerve $|\alpha|$.

Let $f$ be an irreducible one-valued perfect mapping of a paracompactum $X$ onto a paracompactum $Y$.

Fundamental lemma. Under the mapping $f$, every covering $\alpha=\{A_\lambda\}$ of the space $X$ is mapped onto a covering $\beta_\alpha=f\alpha=\{fA_\lambda\}$ of the space $Y$ in a one-to-one manner (in the sense that the correspondence $A_\lambda^\alpha\leftrightarrow fA_\lambda^\alpha$ between the elements of the coverings $\alpha$ and $\beta_\alpha$ is one-to-one). Moreover, for every covering $\beta$ of the space $Y$ one can find at least one covering $\alpha$ of the space $X$ such that $f\alpha=\beta$.

From this lemma it follows:

Theorem 3 (spectral definition of a mapping). An irreducible perfect mapping $f$ of a paracompactum $X$ onto a paracompactum $Y$ determines a passage from the spectrum
$S_X=\{\alpha,\mathfrak{W}_\alpha^{\alpha'}\}$ of the paracompactum $X$ to the spectrum
$S_{XY}=\{\beta_\alpha,\mathfrak{W}_{\beta_\alpha}^{\beta_{\alpha'}}\}$, having as its limit the paracompactum $Y$, in which the $\beta_\alpha$ are directed by the indices $\alpha$; each complex $|\beta_\alpha|$ has the same set of vertices $\dot{\beta}_\alpha=\dot{\alpha}$ as $\alpha$, and contains $\dot{\alpha}$ as a subcomplex; the mapping $f$ also specifies those new simplexes in $\alpha$ by which $|\alpha|$ must be supplemented in order to obtain $|\beta_\alpha|$. The projection $\mathfrak{W}_{\beta_\alpha}^{\beta_{\alpha'}}$ is a simplicial mapping of the complex $|\beta_{\alpha'}|$ onto $|\beta_\alpha|$, given on $|\alpha'|=\dot{\alpha}'$ as the mapping $\mathfrak{W}_\alpha^{\alpha'}$ (and hence is the unique extension of the mapping $\mathfrak{W}_\alpha^{\alpha'}:|\alpha'|\to|\alpha|$ to a simplicial mapping $|\beta_{\alpha'}|\to|\beta_\alpha|$). The mapping $f$ itself is constructed as follows: each maximal thread $x$ of the spectrum $S_X$, being a thread of the spectrum $S_{X,Y}$, is contained in a unique maximal thread $y=fx$ of the spectrum $S_{X,Y}$. From the spectrum $S_Y$ of the paracompactum $Y$, the spectrum $S_{X,Y}$ is obtained by multiplication (see, for example, (5), p. 37) and subsequent weakening of the order.

§ 5. Since $f$ induces a one-to-one mapping of $\dot{\alpha}$ onto $\dot{\beta}_\alpha$ and carries the projection $\mathfrak{W}_\alpha^{\alpha'}$ into $\mathfrak{W}_{\beta_\alpha}^{\beta_{\alpha'}}$, $f$ induces a homeomorphism $\dot{f}$ of the absolute $\dot{X}$ onto the absolute $\dot{Y}$. The mapping $\pi_Y:\dot{Y}\to Y$ can be interpreted as the passage from a thread $\dot{y}=\dot{f}\dot{x}$ of the spectrum $\dot{S}_Y$ to the thread $\dot{x}=\dot{f}^{-1}\dot{y}$, then to the unique maximal thread $x=\pi_X\dot{x}$ of the spectrum $S_X$ containing this thread, and finally to the maximal thread $fx$ of the spectrum $S_{X,Y}$ containing this latter thread. In other words:
\[ \pi_Y=f\pi_X\dot{f}^{-1} \]
and, therefore,

\[ f=\pi_Y \dot f \pi_X^{-1}. \tag{1} \]

Conversely, if a single-valued mapping is representable in the form (1), then it is perfect and irreducible. Thus:

Theorem 4. In order that there exist a single-valued irreducible perfect mapping \(f\) of a paracompactum \(X\) onto a paracompactum \(Y\), it is necessary and sufficient that the absolutes \(\dot X\) and \(\dot Y\) be homeomorphic and that there be a single-valued mapping \(f:X\to Y\), where \(\dot f:\dot X\to \dot Y\) is some homeomorphism between \(\dot X\) and \(\dot Y\).

If this condition is fulfilled, then every irreducible perfect mapping \(f:X\to Y\) is determined by some homeomorphism \(\dot f:\dot X\to \dot Y\) between \(\dot X\) and \(\dot Y\) by formula (1).

Formula (1) may be interpreted as follows: taking \(\dot X=\dot Y\), we may regard \(X\) as the quotient space \(I=\{\pi_X^{-1}x\}\), and \(Y\) as the quotient space \(II=\{\pi_Y^{-1}y\}\) of the same space \(\dot X=\dot Y\). By means of a certain topological mapping of the space \(\dot X\) onto itself we can arrange that each element of the quotient space \(I\) be contained in some element of the quotient space \(II\). Assigning to each element of the first quotient space the element of the second that contains it, we obtain the mapping \(f\).

In conclusion we formulate an easily proved proposition:

Theorem 5. In order that two paracompacta could be mapped (generally speaking, many-valuedly) irreducibly and perfectly onto one another, it is necessary and sufficient that they have homeomorphic absolutes.

Moscow State University
named after M. V. Lomonosov

Received
11 XI 1961

REFERENCES

\({}^{1}\) a) P. Alexandroff, Ann. of Math., 30, 101 (1929); b) A. Kurosch, Comp. math., 2, 471 (1935). \({}^{2}\) V. Ponomarev, DAN, 124, No. 2, 268 (1959); Matem. sborn., 51, 515 (1960). \({}^{3}\) V. Ponomarev, DAN, 132, No. 6, 1269 (1960). \({}^{4}\) P. Alexandrov, V. Ponomarev, Sib. matem. zhurn., 1, 1, 3 (1960). \({}^{5}\) P. Alexandrov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 54 (1959).