MATHEMATICS

B. Pasynkov

ON UNIVERSAL SPACES FOR CERTAIN CLASSES OF SPACES

(Presented by Academician P. S. Aleksandrov on 29 VI 1963)

In this note new classes of spaces will be introduced, generalizing the classes of metric spaces and of spaces with a $\sigma$-star finite base (${}^{1}$), as well as classes of spaces generalizing the classes of paracompact and strongly paracompact spaces. For these new classes of spaces one can construct universal spaces; in doing so one obtains theorems generalizing, for example, K. Morita’s theorem from (${}^{2}$) and Yu. Nagata’s theorems from (${}^{1}$) (see Corollaries 2 and 3).

I. Definition 1. We shall say that a space $^X$ has a $\tau$-refining base if in $X$ there is a refining $^{*}$ system $\nu$ of $\tau$ covers. If, moreover, all the covers in the system $\nu$ are normal ((${}^{11}$), p. 73) and locally finite (respectively, star finite), then we shall say that $X$ has a $\tau$-locally finite (respectively, $\tau$-star finite) base. If, in addition, all covers in the system $\nu$ are $(n+1)$-fold, then we shall say that $X$ has a $\tau$-locally (star) finite $(n+1)$-fold base.

Example 1. All completely regular spaces of weight $\leq \tau$ have a $\tau$-star finite base.

Example 2. Metric ($n$-dimensional in the sense of $\dim$) spaces have an $\aleph_0$-locally finite ($(n+1)$-fold) base.

Example 3. Strongly paracompact ($n$-dimensional) metric spaces have an $\aleph_0$-star finite ($(n+1)$-fold) base.

Example 4. The product of $\tau$ spaces with a $\tau$-refining base again has a $\tau$-refining base.

Example 5. The product of $\tau$ spaces with a $\tau$-locally (star) finite base again has a $\tau$-locally (star) finite base.

Example 6. The product $X$ of $\tau$ spaces with a single-cover $\tau$-star finite base$^{***}$ will again be a space with a single-cover $\tau$-star finite base. It is also clear that $\operatorname{ind} X = 0$.

Example 7. The spaces $B(\mathfrak n,\tau)$, which are the product of $\tau$ spaces $N_\alpha$, $\alpha \in \mathfrak A$, each of which consists of $\mathfrak n$ isolated points, have a $\tau$-star finite base.

For $\tau=\aleph_0$ the spaces $B(\mathfrak n,\aleph_0)$ coincide with the generalized Baire spaces $B(\mathfrak n)$ of weight $\mathfrak n$. Obviously, $\operatorname{ind} B(\mathfrak n,\tau)=0$. For $\tau>\aleph_0$ the spaces $B(\mathfrak n,\tau)$ are completely regular but not normal (${}^{3}$).

$^*$ All spaces considered in the note are assumed (unless the contrary is stated) to be completely regular, all covers open, and all mappings continuous.

$^{}$ A system $\nu$ of covers $\omega_\alpha$, $\alpha\in\mathfrak A$, of a space $X$ is called refining if for every point $x\in X$ and every neighborhood $Ox$ of it there is an index $\alpha=\alpha(x,Ox)$ such that $\operatorname{St}(x,\omega_\alpha)\subset Ox$, where $\operatorname{St}(x,\omega_\alpha)$ denotes the union of those elements of the cover $\omega_\alpha$ which contain the point $x$.

$^{***}$ We note that a single-cover cover is automatically star finite, and a base decomposing into a system of single-cover covers will automatically be refining. Thus, if a space has a base decomposing into a system of $\tau$ single-cover covers, then this space will automatically have a single-cover $\tau$-star finite base.

II. We first consider spaces with a $\tau$-locally finite base, which are a generalization of metric spaces.

Theorem 1. Spaces $X$ of weight $\kappa$ with a $\tau$-locally finite (and $(n+1)$-fold) base, and only they, are homeomorphic to subsets of products of $\tau$ (respectively $n$-dimensional) metric spaces of weight $\kappa$, $\tau \leqslant \kappa$.

Theorem 2. Spaces $X$ of weight $\kappa$ with a $\tau$-locally finite (and $(n+1)$-fold) base, $\tau \leqslant \kappa$, and only they, are homeomorphic to subsets of products of $\tau$ generalized Hilbert spaces $H^\tau$ (respectively spaces $F_n(\Omega)$ from (10), universal for $n$-dimensional metric spaces of weight $\tau$).

Theorem 3. A space $X$ of weight $\kappa$ that is the product of $\tau$ metric spaces $(\tau \leqslant \kappa)$ has a resolving mapping $\tilde f_X$ with $cw(f_X) \leqslant \kappa$ (4) into the Tikhonov cube $I^\tau$.

Theorem 4. For spaces of weight $\kappa$ with a $\tau$-locally finite base $(\tau \leqslant \kappa)$ there exists a universal bicompactum $X(\kappa,\tau)$ of weight $\kappa$, zero-dimensional and openly mapped onto the Tikhonov cube $I^\tau$ ($X(\kappa,\tau)$ is a local product over $I^\tau$ (5)).

Theorem 4 can be generalized:

Theorem 5. Spaces $X$ that are subsets of products of $\tau$ spaces $X_\alpha$, each of which has a resolving mapping $f_\alpha$ with $cw(f_\alpha) \leqslant \kappa$, $\tau \leqslant \kappa$, onto a completely regular space of weight $\tau$, also have a resolving mapping $f$ with $cw(f) \leqslant \tau$ onto a completely regular space of weight $\tau$; that is, the bicompacta $X(\kappa,\tau)$ indicated in Theorem 4 will be universal for these spaces as well.

III. We turn to spaces with a $\tau$-star finite base. It turns out that metric spaces with a base decomposable into the sum of a countable number of star-finite coverings (in (2) they are called spaces with a $\sigma$-star finite base, and in (6)—strongly metrizable spaces) are $\aleph_0$-star finite.

Theorem 6. Completely regular spaces of weight $\kappa$ with a single $\tau$-star finite base, and only they, are homeomorphic to subsets of $B(\kappa,\tau)$.

For $\tau=\aleph_0$ we obtain the known theorem on the universality of generalized Baire spaces $B(\kappa)$ for zero-dimensional in the sense of $\dim$ metric spaces.

Theorem 7. Spaces $X$ of weight $\kappa$ with a $\tau$-star finite ($(n+1)$-fold) base, $\kappa>\tau$, and only they, are homeomorphic to subsets of products of $\tau$ (respectively $n$-dimensional) metric spaces of weight $\leqslant \kappa$, each of which decomposes into the sum of pairwise disjoint open-closed subsets possessing a countable base.

Theorem 8. Every space of weight $\kappa$ with a $\tau$-star finite base is homeomorphic to a subset of the product $B(\kappa,\tau)\times I^\tau$.

Remark 1. It is clear that Theorems 7 and 8 strengthen Theorems 1 and 4 (in the case of spaces with a $\tau$-star finite base).

Corollary 1. Every space $X$ of weight $\kappa$ with a $\tau$-star finite base has such a mapping $\pi$ into $B(\kappa,\tau)$ that the preimage of each point of $B(\kappa,\tau)$ under this mapping has weight $\leqslant \tau$.

Corollary 2 (2). Every space of weight $\kappa$ with an $\aleph_0$-star finite base is homeomorphic to a subset of $B(\kappa)\times I^\infty$ (where $I^\infty$ is the Hilbert cube).

Theorem 9. Every space $X$ of weight $\kappa$ with $\dim X \leqslant n^*$ and with a $\tau$-star finite base is homeomorphic to a subset of the product $F^n\times B(\kappa,\tau)$, where $F^n$ is an $n$-dimensional (in the sense of $\dim$) bicompactum of weight $\tau$, independent of $X$.

Corollary 3 (1). Every metric space of weight $\kappa$ possessing an $\aleph_0$-star finite base is homeomorphic to a subset of: a) the product $B(\kappa)\times M^n$, where $M^n$ is an $n$-dimensional compactum; b) the product $B(\kappa)\times I^{2n+1}$, where $I^{2n+1}$ is the $(2n+1)$-dimensional cube.

* The dimension $\dim$ of a completely regular space is understood in the sense of article (7).

For the proof of the theorems formulated, the following assertion was used:

Theorem 10. A topological (not necessarily completely regular) space (X) if and only if, for its (not necessarily open) cover (\omega), it possesses an (\omega)-mapping into an (n)-dimensional metric space (a generalized polyhedron) (N), when an ((n+1))-fold normal locally finite cover (\gamma) can be inscribed in the cover (\omega). If the cover (\gamma) is star-finite, then the space (N) decomposes into a sum of seven pairwise disjoint open-closed subspaces with countable bases.*

This theorem generalizes Dowker’s theorem from ((^8)), formulated for normal spaces.

IV. Definition 2. We shall call a space (X) (\tau)-normal if all its covers of cardinality (\leq \tau) are normal.

Every normal space is (\omega_0)-normal in the sense that any of its finite covers is normal.

It is clear that (\aleph_0)-normality of a space (X) coincides with its countable paracompactness.

Definition 3. We shall call a cover (\nu) of a space (X) a continuation of a system of sets (\mu), if every element of the system (\mu) belongs to the cover (\nu).

We shall call a space (X) (\tau)-(strongly) paracompact if into every cover (\omega) of the space (X) one can inscribe a cover (\omega'), decomposing into the sum of (\tau) systems, each of which is continued in a locally (star-) finite cover of the space (X)**.

Lemma. Let in a (\tau)-normal space (X) there be given a cover (\omega), decomposing into the sum of (\tau) systems (\omega_\alpha={O_{\alpha\theta}}), (\alpha\in \mathfrak A), (\alpha\theta\in\Theta_\alpha), each of which is continued in a locally (star-) finite cover of the space (X). Then each system (\omega_\alpha) can be continued to such a locally (star-) finite cover (\nu_\alpha={O_{\alpha\theta},V_{\alpha\xi}}), (\alpha\theta\in\Theta_\alpha), (\alpha\xi\in\Xi_\alpha), that for every point (x\in X) there is an index (\alpha) for which
[
x\in \bigcup_{\alpha\theta} O_{\alpha\theta}\setminus \bigcup_{\alpha\xi} V_{\alpha\xi}
= X\setminus \bigcup_{\alpha\xi} V_{\alpha\xi}.
]

From Lemma 1 it follows:

Theorem 11. If a (\tau)-normal and (\tau)-(strongly) paracompact space (X) has a refining system of (\tau) covers, then the space (X) has a (\tau)-locally (star-) finite base.

Besides Theorem 11, Lemma 1 also implies:

Theorem 12. If in a (\tau)-normal space (X) there is a cover (\omega), decomposing into the sum of (\tau) systems (\omega_\alpha), each of which is continued: a) to a locally finite, b) to a star-finite cover, then the space (X) has an (\omega)-mapping into the product of (\tau) spaces: a) metric spaces, b) metric spaces decomposing into a sum of pairwise disjoint open-closed subspaces with countable bases.

In particular, the following is true:

Theorem 13. An (\aleph_0)-strongly paracompact (completely paracompact in the terminology of note ((^6))) space (X), for any of its covers (\omega), has an (\omega)-mapping into the product of a countable number of metric spaces, each of which decomposes into a sum of pairwise disjoint open-closed subspaces with a countable base, i.e. (X), for any of its covers (\omega), has an (\omega)-mapping into (I^\infty\times B(\varkappa)), where (\varkappa) is the weight of (X). If (\dim X=n), then in this case (X), for any of its covers (\omega), has an (\omega)-mapping into the product (M^n\times B(\varkappa)), where (M^n) is an (n)-dimensional compactum (not depending on (X)).

* A mapping (f:X\to R) is called an (\omega)-mapping for the cover (\omega) if for every point (y\in R) there exists a neighborhood (Oy) such that the set (f^{-1}(Oy)) is contained in one of the elements of the cover (\omega).

** Obviously, completely paracompact spaces ((^6)) coincide with (\aleph_0)-strongly paracompact spaces.

Thus, an (\aleph_0)-strongly paracompact space, for each of its covers (\omega), has an (\omega)-mapping onto a space with an (\aleph_0)-star finite base.

From Theorems 10, 13 and from Theorem 6 of paper ((^9)) it follows:

Theorem 14. a) Strongly paracompact spaces (X) (with (\dim X=n)) are spectrally decomposable ((^9)), with respect to the class of (n)-dimensional strongly paracompact metric spaces, each of which is represented in the form of a sum of pairwise disjoint open-and-closed subspaces with a countable base.

b) (\aleph_0)-strongly paracompact spaces (X) (with (\dim X=n)) are spectrally decomposable with respect to the class of (n)-dimensional metric spaces with an (\aleph_0)-star finite base.

Analogous assertions can be formulated for (\tau)-normal (\tau)-(strongly) paracompact spaces (see Theorem 12).

Theorem 1 of ((^9)) can be supplemented by the following theorem:

Theorem 15. A mapping (f:X\to R) of an (n)-dimensional space (X), in the sense of (\dim): a) onto a strongly paracompact metric space; b) onto a space decomposing into a sum of pairwise disjoint open-and-closed subspaces with a countable base; c) onto a metric space with an (\aleph_0)-star finite base, can be represented as the superposition of two mappings (g:X\to S) and (h:S\to R), where (S) is a metric space of weight equal to the weight of (R), (n)-dimensional in the sense of (\dim), and respectively: a) strongly paracompact, b) decomposing into a sum of pairwise disjoint open-and-closed subspaces with a countable base, c) possessing an (\aleph_0)-star finite base (and the set (g(X)) is everywhere dense in (S)).

Let us also note that the space (X) will be (\tau)-(strongly) paracompact if it has a closed and bicompact mapping onto a (\tau)-(strongly) paracompact space; in particular, the space (X) is (\aleph_0)-strongly paracompact (i.e. fully paracompact) if it has a bicompact and closed mapping onto an (\aleph_0)-strongly paracompact space.

Moscow State University
named after M. V. Lomonosov

Received
22 VI 1963

CITED LITERATURE

(^1) J. Nagata, Fund. math., 45, No. 2, 143 (1958).
(^2) K. Morita, Math. Ann., 128, H. 4, 350 (1954).
(^3) A. Stone, Sborn. per. Matematika, 5, No. 5, 1961, p. 3.
(^4) B. Pasynkov, DAN, 144, No. 6, 1217 (1962).
(^5) B. Pasynkov, DAN, 150, No. 1, 40 (1963).
(^6) A. Zarelua, DAN, 141, No. 4, 777 (1961).
(^7) Yu. M. Smirnov, Matem. sborn., 38, No. 3, 283 (1956).
(^8) C. H. Dowker, Bull. Am. Math. Soc., 54, No. 4, 386 (1948).
(^9) B. Pasynkov, DAN, 150, No. 3, 488 (1963).
(^10) J. Nagata, J. reine u. angew. Math., 204, No. 1–4, 132 (1960).
(^11) P. S. Aleksandrov, UMN, 15, No. 2, 25 (1960).