B. PASYNKOV

PARTIAL TOPOLOGICAL PRODUCTS

(Presented by Academician P. S. Aleksandrov on 20 IX 1963)

In the present note partial topological products are introduced, generalizing the ordinary (Tychonoff) topological products (which I shall call complete). Partial products possess many interesting properties and turn out to be well applicable to the construction of universal (in the topological sense) spaces of a given weight and a given dimension, to open-closed zero-dimensional mappings that raise dimension, to finite-to-one closed mappings; to inverse (polyhedral) spectra, to the extension of zero-dimensional mappings to bicompact extensions, etc. All the results of the notes \((^{1,2})\) were obtained with the aid of partial products. In note \((^2)\) partial products are called local products (i.e. in that note the words “local product” must everywhere be replaced by the words “partial product”).

Definition 1. Let a topological space \(X_0\) be given, with an open set \({}_\alpha O_0\) distinguished in it, and a space \(Z_\alpha\). By the partial product \(X_\alpha=P(X_0,Z_\alpha,{}_\alpha O_0)\) of the base \(X_0\) by the fiber \(Z_\alpha\) with respect to the open set \({}_\alpha O_0\) we shall mean a topological space consisting of two disjoint sets \({}_\alpha O_\alpha\) and \(X_\alpha\setminus{}_\alpha O_\alpha\), where the set \(X_\alpha\setminus{}_\alpha O_\alpha\) is homeomorphic to the set \(X_0\setminus{}_\alpha O_0\) (this homeomorphism will be denoted by \({}_0^\alpha\mathfrak F\)), and the set \({}_\alpha O_\alpha\) is homeomorphic to the complete topological product \({}_\alpha O_0\times Z_\alpha\)*; moreover, the natural projection of the complete product
\[ {}_\alpha O_\alpha \equiv {}_\alpha O_0\times Z_\alpha \]
onto the factor \({}_\alpha O_0\) will also be denoted by \({}_0^\alpha\mathfrak F\). As basic open sets in \(X_\alpha\) we shall take: 1) inverse images of open subsets in \(X_0\) under the mapping \({}_0^\alpha\mathfrak F:X_\alpha\to X_0\); 2) open subsets of the set \({}_\alpha O_\alpha\), regarded as the complete topological product \({}_\alpha O_0\times Z_\alpha\).

Definition 2. Let a space \(X_0\) be given, a system of its open subsets \({}_\alpha O_0\), \(\alpha\in\mathfrak A\), and a system of spaces \(Z_\alpha\), \(\alpha\in\mathfrak A\). Then the partial products
\[ X_\alpha=P(X_0,Z_\alpha,{}_\alpha O_0) \]
and the mappings
\[ {}_\alpha^0\mathfrak F:X_\alpha\to X_0,\qquad \alpha\in\mathfrak A, \]
are defined. By the partial topological product
\[ X_{\mathfrak A}=P\bigl(X_0,\{Z_\alpha\},\{{}_\alpha O_0\},\alpha\in\mathfrak A\bigr) \]
of the base \(X_0\) by the system of fibers \(Z_\alpha\) with respect to the system of open sets \({}_\alpha O_0\), \(\alpha\in\mathfrak A\), we shall mean the space whose points are all possible collections
\[ x_{\mathfrak A}=\{x_\alpha\} \]
of points \(x_\alpha\in X_\alpha\) (one point from each \(X_\alpha\)) satisfying, for any two indices \(\alpha'\) and \(\alpha''\in\mathfrak A\), the relation
\[ {}_{\alpha'}^0\mathfrak F(x_{\alpha'})={} _{\alpha''}^0\mathfrak F(x_{\alpha''}). \]
The mapping which assigns to each point \(x_{\mathfrak A}=\{x_\alpha\}\in X_{\mathfrak A}\) the point \(x_\alpha\in X_\alpha\) (the \(\alpha\)-th coordinate of the point \(x_{\mathfrak A}\)) will be denoted by
\[ {}_\alpha^{\mathfrak A}\mathfrak F, \]
i.e.
\[ x_\alpha={}_\alpha^{\mathfrak A}\mathfrak F(x_{\mathfrak A}). \]
We define the topology in \(X_{\mathfrak A}\) so that the collection of all possible sets
\[ \bigl({}_\alpha^{\mathfrak A}\mathfrak F\bigr)^{-1}(V_\alpha),\qquad \alpha\in\mathfrak A, \]
where \(V_\alpha\) is an open set in \(X_\alpha\), forms a subbase.

Remark 1. If \({}_\alpha O_0\equiv X_0\) for all \(\alpha\in\mathfrak A\), then
\[ P\bigl(X_0,\{Z_\alpha\},\{{}_\alpha O_0\},\alpha\in\mathfrak A\bigr)\equiv X_0\times\prod_\alpha Z_\alpha, \]
i.e. partial products generalize complete ones.

\[ \text{* The set }{}_\alpha O_\alpha\text{ will simply be identified with the product }{}_\alpha O_0\times Z_\alpha. \]

Properties of Partial Products

1. The mappings \({}_{0}^{\alpha}\mathfrak F\) and \({}_{\alpha}^{\mathfrak A}\mathfrak F\), \(\alpha \in \mathfrak A\), are continuous, i.e. the mapping
   \(\mathfrak A\mathfrak F = {}_{0}^{\alpha'}\mathfrak F \cdot {}_{\alpha'}^{\alpha''}\mathfrak F \cdot {}_{\alpha''}^{\mathfrak A}\mathfrak F : X_{\mathfrak A} \to X_0\) is continuous, where \(\alpha'\) and \(\alpha''\) are arbitrary indices from \(\mathfrak A\).

2. For every point \(x_{\mathfrak A} \in X_{\mathfrak A}\) there exists a mapping
   \(f_{x_{\mathfrak A}} : X_0 \to X_{\mathfrak A}\) such that
   \(x_{\mathfrak A} \in f_{x_{\mathfrak A}}(X_0)\), the set \(f_{x_{\mathfrak A}}(X_0)\) is closed in \(X_{\mathfrak A}\), and the mapping
   \({}_{0}^{\mathfrak A}\mathfrak F \cdot f_{x_{\mathfrak A}}\) coincides with the identity mapping of the space \(X_0\). In other words, through every point \(x_{\mathfrak A} \in X_{\mathfrak A}\) there passes a secant surface. From this fact follow the relations
   \(\dim X_{\mathfrak A} \geq \dim X_0\),
   \(\operatorname{ind} X_{\mathfrak A} \geq \operatorname{ind} X_0\),
   \(\operatorname{Ind} X_{\mathfrak A} \geq \operatorname{Ind} X_0\).

3. If the base \(X_0\) and all fibers \(Z_\alpha\) are \(T_i\)-spaces, \(i=0,1,2\), then \(X_{\mathfrak A}\) is respectively the same. If \(X_0\) and all \(Z_\alpha\), \(\alpha \in \mathfrak A\), are (completely) regular, then \(X_{\mathfrak A}\) is respectively the same.

4. If \(X_0\) and all \(Z_\alpha\), \(\alpha \in \mathfrak A\), are bicompact (and Hausdorff), then the partial product \(X_{\mathfrak A}\) is the same.

5. \(w(X_{\mathfrak A}) \leq \max\bigl(w(X_0), \max_{\alpha}(w(Z_\alpha)), m(\mathfrak A)\bigr)^*\).

6. If \(\operatorname{ind} Z_\alpha = 0\) for every \(\alpha \in \mathfrak A\), then
   \(\operatorname{ind} X_{\mathfrak A} = \operatorname{ind} X_0\).

7. Suppose that the fiber \(Z_\alpha\) of the partial product
   \(X_\alpha = P(X_0, Z_\alpha, {}_\alpha O_0)\) consists of \(\tau\) isolated points. If the base \(X_0\) is (perfectly) normal, paracompact, metrizable, then \(X_\alpha\) respectively is the same. Moreover, if the set \({}_\alpha O_0\) is of type \(F_\sigma\), then
   \(\dim X_\alpha = \dim X_0\), and if \(X_0\) is perfectly normal, then
   \(\operatorname{Ind} X_\alpha = \operatorname{Ind} X_0\).

8. If all fibers \(Z_\alpha\), \(\alpha \in \mathfrak A\), consist of isolated points, and the system \(\{{}_\alpha O_0\}\), \(\alpha \in \mathfrak A\), is locally finite, then the partial product \(X_{\mathfrak A}\):

a) is locally normal, if the base \(X_0\) is normal; if, in addition, each set \({}_\alpha O_0\) is of type \(F_\sigma\), then
\(\operatorname{loc\,dim} X_{\mathfrak A} \leq \dim X_0\), i.e.
\(\dim X_{\mathfrak A} = \dim X_0\), if \(X_{\mathfrak A}\) is weakly paracompact;

b) is paracompact with \(\dim X_{\mathfrak A} = \dim X_0\), if \(X_0\) is paracompact;

c) is metrizable with \(\dim X_{\mathfrak A} = \dim X_0\), if the base \(X_0\) is metrizable (in this item the system \(\{{}_\alpha O_0\}\), \(\alpha \in \mathfrak A\), may even be assumed locally countable).

1. If all fibers \(Z_\alpha\) of the partial product
   \(X_{\mathfrak A} = P(X_0,\{Z_\alpha\},\{{}_\alpha O_0\}, \alpha \in \mathfrak A)\) are bicompact, then:

a) all mappings \({}_{0}^{\mathfrak A}\mathfrak F\) and \({}_{\alpha}^{\mathfrak A}\mathfrak F\), \(\alpha \in \mathfrak A\), are closed and bicompact, and \(X_{\mathfrak A}\) coincides with the space of such a partition \(\omega_{\mathfrak A}\) of the complete product
\(X_0 \times \prod_{\alpha} Z_\alpha\), that the points
\((x_0',\{z_\alpha'\})\) and \((x_0'',\{z_\alpha''\})\) are contained in one element of the partition \(\omega_{\mathfrak A}\) if and only if
\(x_0' = x_0''\) and \(z_\alpha' = z_\alpha''\) for
\(x_0' = x_0'' \in {}_\alpha O_0\);

b) \(X_{\mathfrak A}\) is (locally) bicompact, finally compact, (weakly, strongly) paracompact, if respectively the base \(X_0\) is so. Moreover, if the base \(X_0\) is paracompact and all fibers \(Z_\alpha\) are bicompact, then
\(\dim X_{\mathfrak A} \leq \dim X_0 + \sum_{\alpha}\dim Z_\alpha\). In particular, if the fibers \(Z_\alpha\) are zero-dimensional bicompacts, then
\(\dim X_{\mathfrak A} = \dim X_0\).

1. The partial product
   \(X_{\mathfrak A} = P(X_0,\{Z_\alpha\},\{{}_\alpha O_0\}, \alpha \in \mathfrak A)\) is the limit of the spectrum
   \(S_{\aleph_0} = \{X_{(\alpha_1\ldots\alpha_s)}, {}_{(\alpha_1\ldots\alpha_k)}^{(\alpha_1\ldots\alpha_s)}\mathfrak F\}\),
   \(\{\alpha \in \mathfrak A\}\), of “elementary” partial products
   \(X_{(\alpha_1\ldots\alpha_s)} = P(X_0,\{Z_{\alpha_i}\},\{{}_{\alpha_i}O_0\}, i=1,\ldots,s)\),
   where
   \({}_{(\alpha_1\ldots\alpha_s)}^{(\alpha_1\ldots\alpha_k)}\mathfrak F\) denotes the naturally arising mapping of the partial product
   \(X_{(\alpha_1,\ldots,\alpha_s)}\) onto the partial product
   \(X_{(\alpha_1\ldots\alpha_k)}\) for
   \(\{\alpha_1,\ldots,\alpha_k\} \subseteq \{\alpha_1,\ldots,\alpha_s\}\)**.

* \(w(X)\) denotes the weight of the space \(X\), \(m(X)\) denotes the cardinality of the set \(X\).

** In the article (2), property 10 was used as the definition of partial products.

Theorem 1. The partial product \(X_{\mathfrak A}=P(I^n,\{Z_\alpha\},\{_\alpha O_0\},\alpha\in\mathfrak A)\), where \(I^n\) denotes the \(n\)-dimensional cube and the system \(\{_\alpha O_0\}\), \(\alpha\in\mathfrak A\), is an arbitrary countable base in \(I^n\), will be a universal space:

a) for all spaces of weight \(\tau\), and only for them, that are at most \(n\)-dimensional in the sense of \(\dim\) metric spaces, if each layer \(Z_\alpha\), \(\alpha\in\mathfrak A\), consists of \(\tau\) isolated points;

b) for all completely regular spaces possessing separating mappings of weight \(\leqslant\tau\) \((^1)\) onto \(n\)-dimensional metric spaces with a countable base, and only for them, if each layer \(Z_\alpha\) is homeomorphic to \(D^\tau\), i.e. is the full product of \(\tau\) spaces consisting of two isolated points. In particular, \(X_{\mathfrak A}\) will be a universal space for all \(n\)-dimensional in the sense of \(\dim\) metric spaces of weight \(\tau\) and for all \(n\)-dimensional in the sense of \(\dim\) bicompacts of weight \(\tau\), zero-dimensionally mapped onto compacta (and also for all the spaces listed in item 2) of Theorem 1 from \((^1)\).

The partial product \(X_{\mathfrak A}\) from items a) and b) satisfies the conditions
\[ w(X_{\mathfrak A})=\tau \quad \text{and} \quad \dim X_{\mathfrak A}=\operatorname{ind}X_{\mathfrak A}=\operatorname{Ind}X_{\mathfrak A}=n. \]

Theorem 2. The partial product \(X_{\mathfrak A}=P(I^\infty,\{Z_\alpha\},\{_\alpha O_0\},\alpha\in\mathfrak A)\), where \(I^\infty\) is the Hilbert brick, and the system \(\{_\alpha O_0\}\) forms a countable base in \(I^\infty\), will be a universal space:

a) for all metric spaces of weight \(\tau\), and only for them, if each layer \(Z_\alpha\), \(\alpha\in\mathfrak A\), consists of \(\tau\) isolated points;

b) for all completely regular spaces possessing a separating mapping of weight \(\tau\) onto a metric space with a countable base, and only for them, if each layer \(Z_\alpha\) is homeomorphic to \(D^\tau\). In particular, \(X_{\mathfrak A}\) will be a universal space for all metric spaces and all bicompacts, zero-dimensionally mapped onto compacta, of weight \(\tau\).

For the product \(X_\alpha\) from items a) and b) the relation
\[ w(X_{\mathfrak A})=\tau \]
will hold.

Remark 2. a) If in item a) of Theorem 1 \(n=0\), then \(X_{\mathfrak A}\) coincides* with the full product of a countable number of spaces consisting of \(\tau\) isolated points, i.e. \(X_{\mathfrak A}\) in this case coincides with the generalized Baire space of weight \(\tau\).

b) For \(\tau=\aleph_0\), item b) of Theorem 1 can be strengthened as follows:

The partial product \(P(I^n,\{Z_\alpha\},\{_\alpha O_0\},\alpha\in\mathfrak A)\), where the system \(\{_\alpha O_0\}\) is a countable base in \(I^n\), and each layer \(Z_\alpha\) consists of two isolated points, will be a universal space for all \(n\)-dimensional metric spaces with a countable base**.

The existence of a universal space among \(n\)-dimensional metric spaces of weight \(\tau\) was first shown in \((^3)\).

Theorem 3. The partial product \(P_\chi^\tau=P(I^\chi,\{Z_\alpha\},\{_\alpha O_0\},\alpha\in\mathfrak A)\), where \(I^\chi\) is the Tikhonov brick of weight \(\chi\), the system \(\{_\alpha O_0\}\), \(\alpha\in\mathfrak A\), forms a base in \(I^\chi\), and each layer \(Z_\alpha\) is homeomorphic to \(D^\tau\), \(\tau\geqslant\chi\), will be a universal space for all completely regular spaces possessing a separating mapping of weight \(\tau\) onto a completely regular space of weight \(\chi\), and only for them. In particular, \(P_\chi^\tau\) will be universal for all bicompacts of weight \(\tau\) possessing a zero-dimensional mapping onto bicompacts of weight \(\chi\), and only for them. \(P_\chi^\tau\) is a bicompactum of weight \(\tau\), zero-dimensionally mapped onto \(I^\chi\).

The following theorem reveals the connections of partial products with locally trivial fiber spaces.

Theorem 4. Let a space \(X_0\) and its covering \(\nu=\{_\alpha O_0\}\), \(\alpha\in\mathfrak A\), be given. The space \(X\) of any such locally trivial fiber space

* Since \(X_0\) consists of one point.
** It is useful to compare this assertion with Theorem 1 from \((^2)\).

\(p: X \to X_0\) with base \(X_0\) and fiber \(Z\), such that each set \(p^{-1}(\alpha O_0)\), \(\alpha \in \mathfrak A\), coincides with the full product \(\alpha O_0 \times Z\) (and the mapping \(p\) on the set \(p^{-1}(\alpha O_0)\) coincides with the projection of the full product \(\alpha O_0 \times Z\) onto the factor \(\alpha O_0\)), has a homeomorphic mapping
\[ f_{\mathfrak A}: X \to P(X_0,\{Z_\alpha\},\{_{\alpha}O_0\},\alpha\in\mathfrak A), \]
where \(p={}_{0}^{\mathfrak A}\mathfrak F \cdot f_{\mathfrak A}\) and \(Z_\alpha \equiv Z\), \(\alpha\in\mathfrak A\).

The following results strengthen and refine Theorem 3 of \((^{1})\) and Theorem 8 of \((^{2})\). In particular, Theorem 5 definitively settles (in the sense of the dimensions of preimages and images) the question of open-closed zero-dimensional mappings of metric spaces.

Theorem 5. Every metric space \(R\) of weight \(\tau\) and with \(\dim R>0\) is an open, closed, bicompact, and zero-dimensional image of a one-dimensional, in the sense of \(\dim\), metric space \(S_R\) of weight \(\tau\), where \(\operatorname{ind} S_R=0\), if \(\operatorname{ind} R=0\).

Theorem 6. a) Every completely regular \(T_1\)-space \(X\) possessing a decomposing mapping of weight \(\tau\) onto a completely regular space of weight \(\chi\) is an open, closed, bicompact, and zero-dimensional image under the mapping \(f_X\) of a completely regular space \(Y_X\) of weight \(\tau\), possessing a decomposing mapping of weight \(\tau\) onto a completely regular space of weight \(\chi\), where \(Y_X\) is a subset of a one-dimensional, in the sense of \(\dim\), bicompactum of weight \(\tau\), and \(w(f_X^{-1}(x)) \leq \chi\) for every point \(x\in X\). If the space \(X\) is paracompact, then the same is true of the space \(Y_X\), and then
\[ \dim Y_X \leq \dim X. \]

b) If in item a) the space \(X\) is strongly paracompact, or finally compact, or bicompact, then, respectively, the space \(Y_X\) will also be the same, and then
\[ \dim Y_X=1. \]

c) In particular, if a bicompactum \(X\) of weight \(\tau\) has a zero-dimensional mapping onto a bicompactum of weight \(\chi\), then \(X\) is an open and zero-dimensional image under the mapping \(f_X\) of a one-dimensional, in the sense of \(\dim\), bicompactum \(Y\) of weight \(\tau\), mapped zero-dimensionally onto a bicompactum of weight \(\chi\), where each set \(f_X^{-1}(x)\), \(x\in X\), has weight \(\chi\).

Note that for \(\chi=\aleph_0\) all the sets \(f_X^{-1}(x)\), \(x\in X\), in items a)—c) will be metrizable, and in items b) and c) the space \(Y_X\) will satisfy the relations
\[ \dim Y_X=\operatorname{ind} Y_X=\operatorname{Ind} Y_X=1 \]
(and in this case, in item b), the space \(X\) may be regarded as completely paracompact \((^{4})\)).

Theorem 7. The following propositions are equivalent:

a) A bicompactum \(X\) of weight \(\tau\) has a zero-dimensional mapping onto a compactum and \(\dim X=n\).

b) The bicompactum \(X\) is the limit of a spectrum
\[ S=\{P_\alpha,{}_{\alpha}^{\beta}\mathfrak F\},\quad \alpha\in\mathfrak A,\quad m(\mathfrak A)=\tau, \]
of \(n\)-dimensional polyhedra \(P_\alpha\), taken in certain triangulations \(K_\alpha\), where each projection \({}_{\alpha}^{\beta}\mathfrak F\) is non-degenerate (i.e. finite-to-one) and simplicial with respect to the triangulation \(K_\beta\) and to some (not necessarily one-fold) barycentric subdivision of the triangulation \(K_\alpha\). The set \(\mathfrak A\) has a minimal index \(0\), by which the \(n\)-dimensional simplex \(E^n\), taken in its natural triangulation \(K_0\), is numbered. Each projection \({}_{\alpha}^{\beta}\mathfrak F\) is represented as a superposition of two-fold projections
\[ {}_{\alpha}^{\alpha_1}\mathfrak F,\ {}_{\alpha_1}^{\alpha_2}\mathfrak F,\ldots,\ {}_{\alpha_s}^{\beta}\mathfrak F. \]
For \(\tau=\aleph_0\), the set \(\mathfrak A\) may be regarded as the natural sequence.

Moscow State University
named after M. V. Lomonosov

Received
18 IX 1963

REFERENCES

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