B. Pasynkov

On a Generalization of Topological Products

(Presented by Academician P. S. Aleksandrov, 1 XII 1962)

All spaces considered below are assumed to be \(T_1\)-spaces, and all mappings continuous. In the present note one important special case of the new concept of a local topological product, generalizing the concept of a topological product, will be introduced and considered, and applications of the new concept will be given. The concept of a local product will be considered in full in a subsequent note.

I. Definition 1 (basic). Let there be given a space \(X_0\)—the base, its open subset \({}_{\alpha}O_0\), and a space \(Z_{\alpha}\)—the fiber, consisting of two isolated points. The local product \(X_{\alpha}=P(X_0,Z_{\alpha},{}_{\alpha}O_0)\) of the base \(X_0\) by the fiber \(Z_{\alpha}\) relative to the open set \({}_{\alpha}O_0\) is defined to be the space of such a decomposition \(\bar g_{\alpha}\) of the topological product
\[ X_0\times Z_{\alpha}=\{(x_0,z_{\alpha}): x_0\in X_0,\ z_{\alpha}\in Z_{\alpha}\}, \]
whose elements are: 1) the individual points \((x_0,z_{\alpha})\in X_0\times Z_{\alpha}\), if \(x_0\in{}_{\alpha}O_0\), and 2) the sets
\[ F_{x_0}=\{(x_0,z_{\alpha}):z_{\alpha}\in Z_{\alpha}\}, \]
if \(x_0\in X_0\setminus{}_{\alpha}O_0\).*

Obviously, the local product coincides with the topological product \(X_0\times Z_{\alpha}\) if \({}_{\alpha}O_0\equiv X_0\).

Denote by \(p_0\) the projection of the product \(X_0\times Z_{\alpha}\) onto the factor \(X_0\). Then there is uniquely defined a mapping
\[ {}^{\alpha}_{0}\mathfrak F:X_{\alpha}\to X_0, \]
satisfying the relation
\[ p_0={}^{\alpha}_{0}\mathfrak F\cdot g_{\alpha}, \]
where \(g_{\alpha}\) is the natural mapping of the space \(X_0\times Z_{\alpha}\) onto the decomposition space \(X_{\alpha}\).

It turns out that the mapping \({}^{\alpha}_{0}\mathfrak F\) is open, closed, two-to-one, piecewise topological \((^1)\), and decomposing \((^2)\). If the space \(X_0\) is respectively Hausdorff, (fully) regular, (perfectly) normal, (locally) bicompact, finally compact, (weakly, strongly) paracompact, then the local product
\[ X_{\alpha}=P(X_0,{}_{\alpha}O_0) \]
will have the same property. Moreover,
\[ \operatorname{ind} X_{\alpha}=\operatorname{ind} X_0; \]
if \(X_0\) is normal, then
\[ \dim X_{\alpha}=\dim X_0; \]
if \(X_0\) is perfectly normal, then
\[ \operatorname{Ind} X_{\alpha}=\operatorname{Ind} X_0. \]

Definition 2. Suppose that in the space \(X_0\) a finite ordered system of open sets
\[ {}_{\alpha_1}O_0,\ldots,{}_{\alpha_s}O_0 \]
is given. According to Definition 1 we construct the local product
\[ X_{\alpha_1}=P(X_0,{}_{\alpha_1}O_0) \]
and the mapping
\[ {}^{\alpha_1}_{0}\mathfrak F. \]
Assuming that the local products
\[ X_{(\alpha_1\ldots\alpha_l)}=P(X_0,{}_{\alpha_1}O_0,\ldots,{}_{\alpha_l}O_0), \]
\(l\le s-1\), and the mappings
\[ {}^{(\alpha_1\ldots\alpha_l)}_{0}\mathfrak F:X_{(\alpha_1\ldots\alpha_l)}\to X_0,\qquad {}^{(\alpha_1\ldots\alpha_l)}_{(\alpha_1\ldots\alpha_k)}\mathfrak F:X_{(\alpha_1\ldots\alpha_l)}\to X_{(\alpha_1\ldots\alpha_k)}, \]
\(k\le l\le s-1\), have already been constructed, we put
\[ X_{(\alpha_1\ldots\alpha_s)}=P(X_0,{}_{\alpha_1}O_0,\ldots,{}_{\alpha_s}O_0) \]
equal to
\[ P\bigl(X_{(\alpha_1\ldots\alpha_{s-1})}, ({}^{(\alpha_1\ldots\alpha_{s-1})}_{0}\mathfrak F)^{-1}({}_{\alpha_s}O_0)\bigr), \]
and the mappings
\[ {}^{(\alpha_1\ldots\alpha_s)}_{0}\mathfrak F:X_{(\alpha_1\ldots\alpha_s)}\to X_0 \]
and
\[ {}^{(\alpha_1\ldots\alpha_s)}_{(\alpha_1\ldots\alpha_k)}\mathfrak F:X_{(\alpha_1\ldots\alpha_s)}\to X_{(\alpha_1\ldots\alpha_k)} \]
are set equal respectively to
\[ {}^{(\alpha_1\ldots\alpha_{s-1})}_{0}\mathfrak F\cdot {}^{(\alpha_1\ldots\alpha_s)}_{(\alpha_1\ldots\alpha_{s-1})}\mathfrak F \]
and
\[ {}^{(\alpha_1\ldots\alpha_{s-1})}_{(\alpha_1\ldots\alpha_k)}\mathfrak F\cdot {}^{(\alpha_1\ldots\alpha_s)}_{(\alpha_1\ldots\alpha_{s-1})}\mathfrak F, \]
where the mapping
\[ {}^{(\alpha_1\ldots\alpha_s)}_{(\alpha_1\ldots\alpha_{s-1})}\mathfrak F: X_{(\alpha_1\ldots\alpha_s)}\to X_{(\alpha_1\ldots\alpha_{s-1})} \]
is constructed in accordance with Definition 1.

It turns out that one may regard
\[ X_{(\alpha_1\ldots\alpha_s)}\equiv X_{(\alpha_{i_1}\ldots\alpha_{i_s})}, \]
where \((\alpha_{i_1}\ldots\alpha_{i_s})\)—

* In an analogous way the local product is also defined when the fiber \(Z_{\alpha}\) is any bicompact space. If \(Z_{\alpha}\) is not a bicompact space, then the definition of a local product is given in a somewhat different form. Since everywhere in this note the fiber \(Z_{\alpha}\) is a simple two-point set, we shall omit the designation of the fiber.

an arbitrary permutation of the set \((\alpha_1\ldots \alpha_s)\); therefore everywhere in what follows the sets of indices \(\alpha\) are understood up to all possible permutations.

Definition 3. Suppose now that in the space \(X_0\) an arbitrary system
\(\mathcal v=\{_{\alpha}O_0\}\), \(\alpha\in\mathfrak A\), of open sets \({}_{\alpha}O_0\) is given. It can be shown that the system of all finite local products

\[ X_{(\alpha_1\ldots \alpha_s)}=P\left(X_0,\{_{\alpha_i}O_0\},\ i=1,\ldots,s\right), \]

connected by the mappings \({}^{(\alpha_1\ldots \alpha_s)}_{(\alpha_1\ldots \alpha_k)}\mathfrak d\) for
\(\{\alpha_1,\ldots,\alpha_k\}\subseteq\{\alpha_1,\ldots,\alpha_s\}\), forms an inverse spectrum

\[ S=\left\{X_{(\alpha_1\ldots \alpha_s)},\ {}^{(\alpha_1\ldots \alpha_s)}_{(\alpha_1\ldots \alpha_k)}\mathfrak d\right\},\quad \alpha\in\mathfrak A, \]

which we shall call conjugate to the system \(\mathcal v\). The limit \(X_{\{\alpha\}}\) of this spectrum we take as the local product \(P\left(X_0,\{_{\alpha}O_0\},\alpha\in\mathfrak A\right)\). It is clear that if \({}_{\alpha}O_0\equiv X_0\) for all \(\alpha\in\mathfrak A\), then

\[ X_{\{\alpha\}}\equiv X_0\times\prod_{\alpha} Z_{\alpha}. \]

The projections \({}^{(\alpha_1\ldots \alpha_s)}_{0}\mathfrak d\) and \({}^{(\alpha_1\ldots \alpha_s)}_{(\alpha_1\ldots \alpha_k)}\mathfrak d\) of the spectrum \(S\) turn out to be finite-to-one, piecewise-topological, open, closed, and decomposing mappings. The projections \(\mathfrak d_{(\alpha_1\ldots \alpha_s)}:X_{\{\alpha\}}\to X_{(\alpha_1\ldots \alpha_s)}\) and \(\mathfrak d_0:X_{\{\alpha\}}\to X_0\) turn out to be open, closed, bicompact, zero-dimensional, and decomposing mappings. If the space \(X_0\) is respectively Hausdorff, (completely) regular, (locally) bicompact, finally compact, (weakly, strongly) paracompact, then so is \(X_{\{\alpha\}}\). Moreover,

\[ \operatorname{ind}X_{\{\alpha\}}=\operatorname{ind}X_0; \]

if \(X_0\) is strongly paracompact, then

\[ \dim X_{\{\alpha\}}=\dim X_0; \]

if \(X_0\) is a perfectly normal bicompactum, then

\[ \operatorname{Ind}X_{\{\alpha\}}=\operatorname{Ind}X_0. \]

Finally,

\[ w\left(X_{\{\alpha\}}\right)\leq \max\left(w(X_0),m(\mathfrak A)\right)^*. \]

II. All the new universal spaces mentioned in the theorems of note \((1)\) are local products; in particular, the bicompacta \(P^{n\tau}\) are local products over \(n\)-dimensional tori \(C^n\) (i.e. \(C^n\) serve as bases for them). It turns out that the Menger universal curve \(M^1\), which is not well suited to the topological product, is a local product over the interval \(I^1\).

Theorem 1. If in the \(n\)-dimensional cube \(I^n\) one takes such a countable base
\(\mathcal v=\{_nO_0\}\), \(n=1,2,\ldots\), that any pair of points \(x'_0\) and \(x''_0\) of \(I^n\) is contained in only a finite number of elements of the base \(\mathcal v\), then the local product

\[ P\left(I^n,\{_nO_0\},n=1,2,\ldots\right)=P^n: \]

1) will be a universal space for all \(n\)-dimensional metric spaces with a countable base; 2) will be locally connected; 3) will not have locally separating points; 4) will be locally universal for all \(n\)-dimensional metric spaces with a countable base, i.e. no open subset of \(P^n\) is embeddable in \(2n\)-dimensional Euclidean space.

From Theorem 1 and from Anderson’s results \((^3)\) it follows:

Theorem 2. The Menger universal curve \(M^1\) is \(P^1\), i.e. \(M^1\) is the local product \(P\left(I^1,\{_nO_0\},n=1,2,\ldots\right)\), where the system \(\{_nO_0\}\) is such a countable base of the interval \(I^1\) that any two of its points are contained in only a finite number of elements of this base.

III. As with topological products, local products are closely connected with the construction of universal spaces. It was already noted above that, for example, the bicompacta \(P^{n\tau}\) are local products. If in the \(n\)-dimensional cube \(I^n\) one takes the same system of open sets \({}_nO_0\), \(n=1,2,\ldots\), as in Theorem 1, and takes the local product

\[ P\left(I^n,\{Z_n\},\{_nO_0\},n=1,2,\ldots\right), \]

where all layers \(Z_n\) are not two-point spaces but \(D^\tau\), i.e. topological products of \(\tau\) copies of a two-point space, then we obtain an \(n\)-dimensional analogue of the Menger curve of weight \(\tau\). The bicompacta obtained will be universal spaces for the same spaces as the bicompacta \(P^{n\tau}\) from \((^1)\), and will have the same properties as \(P^{n\tau}\), with the exception,

\[ \text{* } w(X)\text{ denotes the weight of the space }X,\text{ and }m(A)\text{ the cardinality of the set }A. \]

perhaps, homogeneity. The principal role in the construction of new universal spaces is played by

Theorem 3. In order that the space \(Y\) have a homeomorphic mapping \(f\) into the local product \(P(X_0,\{{}_{\alpha}O_0\}, \alpha\in\mathfrak A)\), it is necessary and sufficient that the space \(Y\) have such a refining mapping \(f_0:Y\to X_0\), for which there exists a base (see (1), p. 1219) consisting of the union of two systems of sets: a) \(\{{}_{\alpha}O_0,{}_{\alpha}O',{}_{\alpha}O''\}, \alpha\in\mathfrak A\), b) \(\{{}_{\beta}V_0,{}_{\beta}V'=f_0^{-1}({}_{\beta}V_0),{}_{\beta}V''=\Lambda\}, \beta\in\mathfrak B\), where the system \(\{{}_{\beta}V_0\}, \beta\in\mathfrak B\), is a base of the space \(X_0\).

Theorems 2, 8, and 10 of (1) can be supplemented by the following assertion.

Theorem 4. Among all: a) completely regular, b) (weakly, strongly) paracompact, c) finally compact, d) bicompact spaces \(X_0\), possessing a refining mapping \(f_0\) with \(cw(f_0)\leq\tau\) (see (1)) into: a) a completely regular, b) a (weakly, strongly) paracompact, c) a finally compact, d) a bicompact space \(X_0\) with \(w(X_0)\leq\tau\), there exists a universal space \(X\) of weight \(\leq\tau\), and \(X\) is a local product \(P(X_0,\{{}_{\alpha}O_0\}, \alpha\in\mathfrak A)\) with respect to some system \(\{{}_{\alpha}O_0\}, \alpha\in\mathfrak A\).

IV. The principal role in applications of local products is played by

Theorem 5. Suppose a mapping \(f_0:Y_0\to X_0\) is given and in the space \(X_0\) there is a system \(\nu=\{{}_{\alpha}O_0\}, \alpha\in\mathfrak A\), of open sets \({}_{\alpha}O_0\), while in the space \(Y_0\) there is a system \(\mu=\{{}_{\alpha}V_0\}, \alpha\in\mathfrak A\), of open sets \({}_{\alpha}V_0\), with \({}_{\alpha}V_0\supset f_0^{-1}({}_{\alpha}O_0)\). It turns out that there exists a mapping
\[ f:Y_{\{\alpha\}}=P(Y_0,\{{}_{\alpha}V_0\}, \alpha\in\mathfrak A)\to X_{\{\alpha\}}=P(X_0,\{{}_{\alpha}O_0\}, \alpha\in\mathfrak A) \]
such that: 1) \(f_0\cdot\pi_0=\mathfrak E_0\cdot f\), where \(\pi_0\) and \(\mathfrak E_0\) (defined earlier) are the mappings of the local products \(Y_{\{\alpha\}}\) and \(X_{\{\alpha\}}\) onto the bases \(Y_0\) and \(X_0\); 2) \(f\cdot\pi_0^{-1}=\mathfrak E_0^{-1}\cdot f_0\), i.e. \(f(Y_{\{\alpha\}})=X_{\{\alpha\}}\) when \(f_0(Y_0)=X_0\); 3) \(\operatorname{ind} f=\operatorname{ind} f_0\); 4) if the mapping \(f_0\) is closed and bicompact, then the mapping \(f\) is closed and bicompact; 5) if \({}_{\alpha}V_0=f_0^{-1}({}_{\alpha}O_0)\) for all \(\alpha\in\mathfrak A\) and the mapping \(f_0\) is open, then the mapping \(f\) is open as well; 6) if \({}_{\alpha}V_0=f_0^{-1}({}_{\alpha}O_0)\) for all \(\alpha\in\mathfrak A\) and \(\mathfrak E_0(x)=x_0\), then the sets \(f_0^{-1}(x_0)\) and \(f^{-1}(x)\) are homeomorphic to one another by means of the mapping \(\pi_0\). In particular, if the mapping \(f_0\) is: a) \((k+1)\)-fold, \(k=1,2,\ldots\), b) finite (countably) multiple, c) \(n\)-dimensional in the sense \(\dim(\operatorname{ind})\), \(n=0,1,2,\ldots\), then the mapping \(f\) will be the same; if all sets \(f_0^{-1}(x_0)\), \(x_0\in X_0\), are homeomorphic to one another, then also all sets \(f^{-1}(x)\), \(x\in X_{\{\alpha\}}\), are homeomorphic to one another.

Corollary 1. If \(X_0\) is a dyadic bicompactum, then any local product over \(X_0\) is a dyadic bicompactum.

As an application of Theorem 5 we obtain the following two assertions.

Theorem 6. If the space \(X_0\) is a \((k+1)\)-fold and closed image of a completely regular space \(Y_0\) with \(\operatorname{ind}Y_0=0\), then any local product \(X_{\{\alpha\}}=P(X_0,\{{}_{\alpha}O_0\},\alpha\in\mathfrak A)\) over \(X_0\) will be a \((k+1)\)-fold and closed image of a completely regular space \(Y_{\{\alpha\}}\) with \(\operatorname{ind}Y_{\{\alpha\}}=0\), and if the space \(Y_0\): a) is bicompact, b) finally compact, c) (weakly, strongly) paracompact, then \(Y_{\{\alpha\}}\) will be the same; i.e. if \(X_0\) is a perfectly \(k\)-dimensional paracompactum (4,5), then so are all local products \(X_{\{\alpha\}}\) over \(X_0\), i.e. in this case \(\dim X_{\{\alpha\}}=\operatorname{ind}X_{\{\alpha\}}=\operatorname{Ind}X_{\{\alpha\}}=k\).

Theorem 7. If the space \(A\) has a refining mapping into the space \(X_0\), which is a \((k+1)\)-fold and closed image of a completely regular space \(Y_0\) with \(\operatorname{ind}Y_0=0\)*, then the space \(A\) will also be a \((k+1)\)-fold and closed image of a completely regular space \(B\) with \(\operatorname{ind}B=0\), and if the space \(A\) is (weakly, strongly) paracompact (finally compact, bicompact), then the space

* For example, the space \(Y_0\) may be a metric space with \(\dim Y_0=k\).

space \(B\), i.e., if \(A\) is strongly paracompact, then it will be perfectly \(k\)-dimensional, i.e. \(\dim A=\operatorname{ind} A=\operatorname{Ind} A\).

V. Theorem 3 from \({}^{(1)}\) is also an application of Theorem 5. The results of Theorem 3 from \({}^{(1)}\) can be strengthened:

Theorem 8. 1) Every compactum is an open and zero-dimensional image of a one-dimensional compactum.

2) Every bicompactum \(X\) of weight \(\tau\) is an open and zero-dimensional image of a one-dimensional, in the sense of \(\dim\), bicompactum \(Y\) of weight \(\tau\). If \(X\) has a zero-dimensional mapping onto a compactum, then \(Y\) also has a zero-dimensional mapping onto a compactum, and then \(\dim Y=\operatorname{ind} Y=\operatorname{Ind} Y=1\).

3) Every: a) completely regular, b) (weakly, strongly) paracompact, c) finally compact space \(X\) of weight \(\tau\) is an open, closed, bicompact, and zero-dimensional image of: a) a completely regular, b) a (weakly, strongly) paracompact space \(Y\) of weight \(\tau\), which is a subspace of a one-dimensional, in the sense of \(\dim\), bicompactum, i.e. if \(X\) is a strongly paracompact (finally compact) space, then \(\dim Y=1\). If the space \(X\) has a refining mapping onto a metric space, then the space \(Y\) also has a refining mapping onto a metric space, and in this case \(\operatorname{ind} Y=1\), i.e. if \(X\) is a strongly paracompact space, then \(\dim Y=\operatorname{ind} Y=\operatorname{Ind} Y=1\).

VI. Local products make it possible to represent refining mappings (in a certain sense) as superpositions of two-fold mappings:

Theorem 9. A mapping \(f_{0}\) of a space \(Y\) onto a space \(Y_{0}\) will be refining if and only if the space \(Y\) is an everywhere dense subset of the limit \(\bar{Y}\) of such a spectrum \(S_{Y}=\{Y_{\alpha}, \mathfrak{F}_{\alpha}^{\beta}\}\), \(\alpha\in\mathfrak{A}\), that: 1) the projections \(\mathfrak{F}_{\alpha}^{\beta}\) are finite-to-one, piecewise-topological, refining mappings “onto”; 2) the space \(Y_{0}\) is a minimal element in the spectrum \(S_{Y}\), i.e. \(0<a\) for every index \(\alpha\in\mathfrak{A}\); 3) the projection \(\mathfrak{F}_{0}:\bar{Y}\to Y_{0}\) coincides on the set \(Y\subseteq\bar{Y}\) with the mapping \(f_{0}\); 4) every projection \(\mathfrak{F}_{0}^{\alpha}:Y_{\alpha}\to Y_{0}\) is represented as a superposition of a finite number of two-fold projections \(\mathfrak{F}_{\alpha_{1}}^{\alpha}, \mathfrak{F}_{\alpha_{2}}^{\alpha_{1}},\ldots,\mathfrak{F}_{\alpha_{s}}^{\alpha_{s-1}}, \mathfrak{F}_{0}^{\alpha_{s}}\), \(s=s(\alpha)\); 5) then and only then \(Y\equiv\bar{Y}\), when the mapping \(f_{0}\) is bicompact.

In particular, Theorem 9 gives a characterization of zero-dimensional mappings of bicompacta and locally bicompact spaces. For example, for compacta we have the following theorem:

Theorem 10. A mapping \(f_{0}\) of a compactum \(\Phi\) onto a compactum \(\Phi_{0}\) is zero-dimensional if and only if \(\Phi\) is the limit of a spectrum \(S=\{\Phi_{n},\mathfrak{F}_{n}^{m}\}\), \(n=0,1,2,\ldots\), where each projection \(\mathfrak{F}_{n}^{\,n+1}\), \(n=0,1,2,\ldots\), is a two-fold and piecewise-topological mapping “onto” and \(\mathfrak{F}_{0}\equiv f_{0}\).

Theorem 11. On the local product \(X_{\alpha}=P(X_{0},\{_{\alpha}O_{0}\},\alpha\in\mathfrak{A})\) there acts a bicompact zero-dimensional commutative group \(D^{\tau}\), which is the direct product of \(\tau\) groups of the second order, where \(\tau=m(\mathfrak{A})\), and the orbit space of the space \(X_{\{\alpha\}}\) under the action of the group \(D^{\tau}\) coincides with the space \(X_{0}\), while the mapping \(\mathfrak{F}_{0}:X_{\{\alpha\}}\to X_{0}\) coincides with the natural mapping of \(X_{\{\alpha\}}\) onto the orbit space. If the space \(X_{0}\) is homogeneous, and the system \(\nu=\{_{\alpha}O_{0}\}, \alpha\in\mathfrak{A}\), is such that together with the set \({}_{\alpha}O_{0}\) the system \(\nu\) also contains all sets \(g({}_{\alpha}O_{0})\), \(g\in G\), where \(G\) is a group acting on \(X_{0}\), then the space \(X_{\{\alpha\}}\) is also homogeneous.

Received
24 XII 1962

REFERENCES

\({}^{1}\) B. Pasynkov, DAN, 144, No. 6 (1962). \({}^{2}\) A. Zarelua, DAN, 144, No. 4 (1962). \({}^{3}\) R. D. Anderson, Ann. Math., 68, No. 1 (1958). \({}^{4}\) P. Aleksandrov, V. Ponomarev, Siberian Math. Journal, 1, No. 1 (1960). \({}^{5}\) V. Ponomarev, DAN, 144, No. 5 (1962).