B. PASYNKOV

ON $\omega$-MAPPINGS AND INVERSE SPECTRA

(Presented by Academician P. S. Aleksandrov on 17 XII 1962)

All spaces considered below are assumed to be Hausdorff, and the mappings continuous.

Theorem 1. If there is a countable system of mappings $f_i : X \to R_i$, $i = 1, 2, \ldots$, of a normal space $X$ with $\dim X = r$ onto metric spaces (respectively, onto metric spaces with a countable base) $R_i$, then there exists a metric space (respectively, a metric space with a countable base) $S$ with $\dim S \leq r$ and mappings $g : X \to S$ and $h_i : S \to R_i$ such that $h_i \cdot g = f_i$, $i = 1, 2, \ldots$.

Remark 1. Theorem 1 generalizes Lemma 4 of (¹), for if $X$ is bicompact, then $S$ is compact.

Theorem 2. If a normal space $X$ with $\dim X = r$ is the limit of some spectrum of metric spaces (respectively, of metric spaces with a countable base), then $X$ is also the limit of a spectrum of $r$-dimensional, in the sense of $\dim$, metric spaces (metric spaces with a countable base).

Definition 1. A space $X$ will be called spectrally decomposable with respect to a class of spaces $\mathfrak M$ (a spectral $\mathfrak M$-space) if $X$ is the limit of some inverse spectrum $S = {X_\alpha, \mathfrak F_\alpha^\beta}$, $\alpha \in \mathfrak A$, where all $X_\alpha$ belong to the class $\mathfrak M$.

Definition 2. A class of spaces $\mathfrak M$ will be called: 1) monotone, if from $Y \in \mathfrak M$ and from $A \subset Y$ it follows that $A \in \mathfrak M$; 2) closed with respect to finite multiplication, if from $Y$ and $Z \in \mathfrak M$ it follows that $Y \times Z \in \mathfrak M$; 3) sufficiently broad, if $\mathfrak M$ contains all metric spaces with a countable base.

A class of spaces $\mathfrak M$ satisfying conditions 1), 2) or, respectively, conditions 1), 2), 3), will be called a class of type (1—2) or, respectively, a class of type (1—3).

Examples of classes of type (1—3) are: the classes of spaces with the first axiom of countability, spaces with a countable base, metric spaces, strongly metrizable metric spaces (²), metric spaces with a countable base.

Theorem 3. If a Hausdorff space $X$ is the limit of a spectrum $S = {X_\alpha, \mathfrak F_\alpha^\beta}$, $\alpha \in \mathfrak A$, of Hausdorff spaces $X_\alpha$ with projections $\mathfrak F_\alpha^\beta$ that are, in general, mappings “onto,” then for every Hausdorff one-point extension $\widetilde X = X \cup x_0$ of the space $X$ there exists an index $\alpha_0 \in \mathfrak A$ such that the projection $\mathfrak F_{\alpha_0} : X \to X_{\alpha_0}$ cannot be extended to a continuous mapping of the extension $\widetilde X$ into the space $X_{\alpha_0}$.

Definition 3. A mapping $f : X \to Y$ with respect to some covering $\omega$ of the space $X$ is called: 1) an $\omega$-mapping (respectively, a finite $\omega$-mapping, a countable $\omega$-mapping) if for every point $y \in Y$ there exists a neighborhood $O_y$ such that the set $f^{-1}(O_y)$ is wholly contained in one element (in a finite number of elements, in a countable number of elements) of the cover-

(\omega); 2) a weak (\omega)-mapping (respectively, a finite weak (\omega)-mapping, a countable weak (\omega)-mapping) if for each point (y \in Y) the set (f^{-1}(y)) is wholly contained in one element (respectively in a finite, in a countable number of elements) of the cover (\omega).

Remark 2. Every bicompact (finally compact) mapping (f : X \to Y) is a finite (countable) weak (\omega)-mapping for any cover (\omega) of the space (X); if, moreover, the mapping (f) is closed, then it is a finite (countable) (\omega)-mapping for any cover (\omega) of the space (X).

Theorem 4. In order that a (T_1)-space (X) have a homeomorphic mapping onto an everywhere dense subset of the limit of some spectrum
(S={X_\alpha,\mathfrak{F}\alpha^\beta},\ \alpha \in \mathfrak{A}), of spaces (X\alpha) belonging to a class (\mathfrak{M}) of type ((1—2)), it is necessary and sufficient that for every point-binary cover* (\omega) of the space (X) there exist an (\omega)-mapping
(f_\omega : X \to X_\omega), where (X_\omega \in \mathfrak{M}).

Theorem 5. In order that a (T_1)-space (X) have a one-to-one and continuous mapping onto the limit of some spectrum
(S={X_\alpha,\mathfrak{F}\alpha^\beta},\ \alpha \in \mathfrak{A}), of spaces (X\alpha) from some class of Hausdorff spaces (\mathfrak{M}) of type ((1—2)), it is sufficient that for every cover (\omega) of the space (X) there exist a weak (\omega)-mapping
(g_\omega : X \to Y_\omega), where (Y_\omega \in \mathfrak{M}).

Theorem 6. In order that a (T_1)-space (X) be a spectral (\mathfrak{M})-space, where (\mathfrak{M}) is a class of Hausdorff spaces of type ((1—2)), it is sufficient that two conditions be fulfilled: 1) for every point-binary cover (\omega) of its own, the space (X) have an (\omega)-mapping (f_\omega : X \to X_\omega), where (X_\omega \in \mathfrak{M}); 2) for every cover (\omega) of its own, the space (X) have a weak (\omega)-mapping
(g_\omega : X \to Y_\omega), where (Y_\omega \in \mathfrak{M}).

In particular, a (T_1)-space (X) will be a spectral (\mathfrak{M})-space, where (\mathfrak{M}) is a class of Hausdorff spaces of type ((1—2)), if for every cover (\omega) of its own the space (X) has an (\omega)-mapping into some space (X_\omega \in \mathfrak{M}).

Since every: a) paracompact, b) strongly paracompact, c) regular finally compact, d) bicompact Hausdorff space (X), for any cover (\omega) of its own, has an (\omega)-mapping respectively onto: a) a metric space, b) a strongly metrizable metric space, c) a metric space with a countable base, d) a compact metric space (see ((^{3,4}))), it follows that

Theorem 7. a) Every paracompactum is spectrally decomposable with respect to the class of metric spaces; b) every strongly paracompact Hausdorff space is spectrally decomposable with respect to the class of strongly metrizable metric spaces; c) every regular finally compact space is spectrally decomposable with respect to the class of metric spaces with a countable base**.

From Theorems 2 and 7 it follows that

Theorem 8. a) Every paracompactum that is (r)-dimensional in the sense of (\dim) is the limit of some spectrum of (r)-dimensional, in the sense of (\dim), metric spaces; b) every (r)-dimensional, in the sense of (\dim), regular finally compact space is the limit of some spectrum of (r)-dimensional metric spaces with a countable base.

As a consequence of Theorem 8 we obtain Theorem 1 from ((^1)).

Theorem 9. In order that a strongly paracompact space (X) have (\dim X \leq r), it is necessary and sufficient that the space (X) be spectrally decomposable with respect to the class of (r)-dimensional, in the sense of (\dim), metric spaces***.

* A point-binary cover is any cover of the form ({Ox,\ X\setminus x}), where (Ox) is a neighborhood of the point (x).

** Parts a) and c) of this theorem were also proved by V. Ponomarev.

*** The sufficiency of the formulated condition was also proved by Yu. M. Smirnov.

Theorem 10. In order that a Hausdorff (respectively, completely regular) space (X) be spectrally decomposable with respect to the class of Hausdorff spaces (\mathfrak M) of type (1–2), it is necessary and sufficient that: a) for each of its point-binary covers (\omega), the space (X) have an (\omega)-map (f_\omega:X\to X_\omega), where (X_\omega\in\mathfrak M); b) for any of its Hausdorff (respectively, completely regular) one-point extensions (X\cup x_0), the space (X) have such a map (f_{x_0}:X\to X_{x_0}), (X_{x_0}\in\mathfrak M), which cannot be extended to a continuous map of the extension (X\cup x_0) into the space (X_{x_0}).

Theorem 11. In order that a completely regular space (X) be spectrally decomposable with respect to the class of metric spaces with a countable base, it is necessary and sufficient that one of the following conditions be fulfilled: a) for any point (x_0\in \beta X\setminus X^) there exists a map (f_{x_0}) of the space (X) onto a metric space with a countable base (X_{x_0}), which cannot be extended to a continuous map of the set (X\cup x_0\subseteq \beta X) into the space (X_{x_0}); b) every point (x_0\in\beta X\setminus X) is a set of type (G_\delta) (has a countable pseudocharacter) in the set (X\cup x_0\subseteq\beta X); c) for any point (x_0\in\beta X\setminus X) there exists a countable normal* star-finite cover of the space (X), the closures (in (\beta X)) of whose elements do not meet the point (x_0).

Theorem 12. A normal space (X) is spectrally decomposable with respect to the class of metric spaces with a countable base if and only if it has such a map (f) onto some metric space with a countable base (Y) that every set (f^{-1}(y)), (y\in Y), is spectrally decomposable with respect to the class of metric spaces with a countable base.

Theorem 13. In order that a completely regular space (X) be spectrally decomposable with respect to the class of metric spaces (respectively, strongly metrizable metric spaces, metric spaces with a countable base), it is sufficient that for each of its covers (\omega) the space (X) have a countably weak (\omega)-map onto some space of the corresponding class.

In particular, if the space (X) has a bicompact or even finally compact map onto some metric space (respectively, onto a strongly metrizable metric space, a metric space with a countable base), then it is spectrally decomposable with respect to the corresponding class of metric spaces.***

Theorem 14. a) A completely regular space (X) is spectrally decomposable with respect to the class of metric spaces if and only if for any point (x_0\in\beta X\setminus X) there exists such a map (f) of the space (X) onto a metric space (X_{x_0}) which cannot be extended to a continuous map of the set (X\cup x_0\subseteq\beta X) onto the space (X_{x_0}); b) a completely regular space (X) is spectrally decomposable with respect to the class of metric spaces if and only if for any point (x_0\in\beta X\setminus X) there exists such a normal locally finite in (X) cover whose closures of elements in (\beta X) do not meet the point (x_0).

Theorem 15. The classes of spectrally paracompact (spectrally finally compact) and spectrally metric spaces (spectrally metric spaces with a countable base) coincide****, and coincide

* By (\beta X) everywhere is meant the maximal (Stone–Čech) bicompact extension of the space (X).

** See (5), footnote on p. 73.

*** We note that a finally compact space has a finally compact map simply to a point.

**** That is (see Definition 1), if a space (X) is the limit of a spectrum of paracompact (finally compact) spaces, then it is the limit of a spectrum of metric spaces (with a countable base).

with the class of spaces that are closed subsets of direct products of paracompact (finally compact), or, what is the same, metrizable spaces (with a countable base).

If a spectrally paracompact space (X) is normal and (\dim X=r), then (X) may be regarded as a closed subset of a product of (r)-dimensional metrizable spaces.

Theorem 16. a) A completely regular space (X) is functionally closed if and only if it is spectrally finally compact.
b) A completely regular space (X) is complete in the sense of Dieudonné if and only if it is spectrally paracompact.

We shall call a spectrally paracompact (spectrally finally compact) space (\widetilde X) a spectrally paracompact (spectrally finally compact) extension of the space (X), if (X) is everywhere dense in (\widetilde X) and in (\widetilde X) there is no spectrally paracompact (spectrally finally compact) subset (X') such that
[
X \subseteq X' \subset \widetilde X .
]

Theorem 17. a) The completion of a completely regular space (X) with respect to its maximal uniform structure coincides with that spectrally paracompact extension (\mu X) of the space (X) to which every mapping of the space (X) into a metrizable (spectrally metrizable) space extends.
b) The Hewitt (i.e., maximal functionally closed) extension (\upsilon X) (({}^{5}),) p. 72) of a completely regular space (X) coincides with that spectrally finally compact extension of the space (X) to which every mapping of the space (X) into a metrizable space with a countable base (into a spectrally finally compact space) extends.

Received
12 XII 1962

REFERENCES CITED

(^{1}) S. Mardešić, Illinois J. Math., 4, No. 2 (1960).
(^{2}) A. Zarelua, DAN, 141, No. 4 (1961).
(^{3}) C. H. Dowker, Ann. of Math., 51 (1950).
(^{4}) V. Ponomarev, DAN, 141, No. 3 (1961).
(^{5}) P. S. Aleksandrov, UMN, 15, No. 2 (1960).