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UDC 513.83
MATHEMATICS
N. S. LASHNEV
ON PERFECT IRREDUCIBLE MAPPINGS OF COMPLETELY REGULAR SPACES
(Presented by Academician P. S. Aleksandrov on 17 I 1969)
All mappings are assumed to be continuous.
Definition 1. Let there be an inverse system
\(\Sigma = \{X_\alpha, \pi_{\alpha_2}^{\alpha_1}\}\) \((\alpha \in A)\) of topological spaces \(X_\alpha\) with continuous mappings \(\pi_{\alpha_2}^{\alpha_1}\), constructed on the directed set \(A=\{\alpha\}\). Suppose that in this set there is a minimal element \(\alpha_0\), for which \(\alpha \geq \alpha_0\) for every \(\alpha\). Then for every space \(X_\alpha\) \((\alpha \in A)\) there exists a mapping onto \(X_{\alpha_0}\). The limit of the system \(X\) can also be naturally mapped onto \(X_{\alpha_0}\). We shall say that the mapping \(\pi: X \to X_{\alpha_0}\) is decomposed into a system of mappings \(\pi_{\alpha_0}^{\alpha}: X_\alpha \to X_{\alpha_0}\).
Definition 2. A mapping \(f: X \to Y\) will be called elementary if in \(X\) there is an open everywhere dense set \(U\) on which \(f\) is a homeomorphism.
§ 1. The extension \(d_XY\) and its properties
Let \(X\) and \(Y\) be locally bicompact spaces; \(f: X \to Y\) a perfect irreducible mapping; \(bX\) a bicompact extension of the space \(X\); \(\alpha\) a proximity space on \(X\) generated by the extension \(bX\). Define on \(Y\) a proximity space \(\beta\): \(A \bar{\beta} B\) if and only if
\[ (f^{-1}A)\alpha(f^{-1}-B). \]
All the axioms of a proximity space are easily verified.
The proximity space \(\beta\) corresponds to a bicompact extension of the space \(Y\), which we shall denote by \(d_{bX}Y\), or, when it is clear which \(bX\) and \(f\) are meant, simply by \(dY\).
We establish some properties of the extension.
Theorem 1. If the mapping \(f: X \to bY\) can be extended to a mapping \(g: bX \to bY\), then there exists a mapping \(h: d_{bX}Y \to bY\), identical on \(Y\).
The proof follows easily from the definition of \(d_{bX}Y\).
Theorem 2. Let \(g\) be the embedding \(Y \to d_{bX}Y\); then the mapping \(fg: X \to dY\) can be extended to a mapping \(bX \to dY\).
Part of the proof of Theorem 2. Let \(x \in bX\), and let \(\delta\) be the system of all neighborhoods of the point \(x\): \(\delta=\{S_\alpha\}_{\alpha \in A}\). Put \(T_\alpha = f^{\#}(S_\alpha \cap X)\)—this is a nonempty open set by virtue of the irreducibility and closedness of the mapping \(f\). If there exists a finite family of sets \(T_{\alpha_1}, T_{\alpha_2}, T_{\alpha_3}, \ldots, T_{\alpha_n}\), then
\[ f^{-1}\left(\bigcap_{k=1}^{n} T_{\alpha_k}\right) \subset X \cap \left(\bigcap_{k=1}^{n} S_{\alpha_k}\right). \]
From the centeredness of the system \(\delta\) follows the centeredness of the system \(\zeta=\{T_\alpha\}_{\alpha \in A}\). Since \(dY\) is bicompact, \(\zeta\) has a cluster point in \(dY\). We prove that this point is unique.
- Suppose that this is not so: \(y_1 \in [\zeta]\), \(y_2 \subset [\zeta]\), \(y_1 \ne y_2\). (By \([\zeta]\) we denote the set of all cluster points of the filter \(\zeta\).)
-
At the points \(y_1\) and \(y_2\) choose neighborhoods \(U_1\) and \(U_2\) with disjoint closures in \(d_bX\).
-
\(U_1 \cap Y\) and \(U_2 \cap Y\) are far in the proximity space \(\beta\).
-
Denote \(W_1=f^{-1}(U_1)\), \(W_2=f^{-1}(U_2)\).
-
\(W_1 \,\alpha\, W_2\), according to the definition of the proximity \(\beta\).
-
Hence it follows that \([W_1]_{bX}\cap[W_2]_{bX}=\Lambda\).
-
\(U_1\cap T=\Lambda\), \(U_2\cap T=\Lambda\) for every \(T\in\xi\).
-
Hence, and from the definition of the filter \(\xi\), it follows that \(W_1\cap S=\Lambda\), \(W_2\cap S=\Lambda\) for every \(S\in\delta\).
-
Since \(\delta\) is the filter of all neighborhoods of the point \(x\), \([W_1]\ni x\), \([W_2]\ni x\).
-
Items 9 and 6 contradict each other, as was required to prove.
Assign to each point \(x\in X\) the unique limit point \(y\) of the filter \(\zeta(x)\). The mapping \(g\) thus constructed will be the desired mapping \(bX\to dY\).
§ 2. Decomposition of an irreducible mapping into elementary mappings. Let \(f:X\to Y\) be an irreducible mapping of the bicompactum \(X\) onto the bicompactum \(Y\). Let \(\mathfrak B\) be a base in \(X\) of cardinality \(\tau\); \(U_1\) and \(U_2\) are elements of the base; \([U_1]\cap[U_2]=\Lambda\). Put
\[
F(U_1,U_2)=f[U_1]\cap f[U_2].
\]
Since the mapping \(f\) is irreducible, \(F\) is nowhere dense. \(Y\setminus F\) is a locally bicompact space; it is a perfect irreducible image of its full preimage \(P\subset X\). By Theorem 2 there exists a bicompact extension \(d_X(Y\setminus F)\) with the natural mapping
\[
g:X\to d_X(Y\setminus F).
\]
Lemma. \(g(U_1)\cap g(U_2)=\Lambda\).
Proof of the lemma. The open sets \(U_1\cap f^{-1}(Y\setminus F)\) and \(U_2\cap f^{-1}(Y\setminus F)\) have disjoint closures in \(X\). Their marked kernels are far in \(\alpha\); consequently, the images \(V_1\) and \(V_2\) of these kernels in \(Y\setminus F\) are far in the proximity space \(\beta\). Hence it follows that
\[
[V_1]_d\cap[V_2]_d=\Lambda,
\]
where \(d=d(Y\setminus F)\).
It is easy to verify that
\[
g(U_i)\subseteq [V_i]_{d(Y\setminus F)}\quad (i=1,2),
\]
as was required to prove.
Since \(\mathfrak B\) is a base of the bicompactum \(X\) of cardinality \(\tau\), for any two points \(x_1\) and \(x_2\) of \(X\) one can choose neighborhoods \(U_1\) and \(U_2\) with disjoint closures. It follows from the lemma that the mapping
\[
g:X\to d_X\bigl[Y\setminus F(U_1,U_2)\bigr]
\]
sends the points \(x_1\) and \(x_2\) to distinct points \(r_1\) and \(r_2\) in the space \(d_X\).
Let \(\eta\) be the system of all sets \(F(U_1,U_2)\), where \(U_1,U_2\in\mathfrak B\), and let \(\vartheta\) be the system of all finite unions of sets from \(\eta\). Obviously, the cardinality of the system \(\vartheta\) is equal to \(\tau\). The system of all extensions \(d_X(Y\setminus P)\) (\(P\in\vartheta\)) forms an inverse spectrum \(\Sigma(\mathfrak B)\) with natural projections
\[
d_X(Y\setminus P_1)\to d(Y\setminus P_2),
\]
when \(P_1\subseteq P_2\). (For the construction of the spectrum see \((^2)\).) Denote the limit of the spectrum \(\Sigma(\mathfrak B)\) by \(Y\).
Theorem 3. \(X=\lim\Sigma(\mathfrak B)\).
Proof. Consider the system of mappings
\[
\{g_\lambda:X\to d_X(Y\setminus F_\lambda)\}.
\]
It generates a mapping
\[
\{g:X\to d_X(Y\setminus F_\lambda)\}.
\]
For any two points of \(X\) there is a mapping that sends them to distinct points in \(d_X(Y\setminus F_{\lambda(x_1,x_2)})\). From the definition of the limit of a spectrum it follows at once that \(\tilde g(x_1)\ne \tilde g(x_2)\). This means that the mapping \(g\) is one-to-one, i.e. a homeomorphism, as was required to prove.
From Theorem 3 it follows:
Theorem 4. An irreducible mapping of a bicompactum \(X\) of weight \(\tau\) onto a bicompactum \(Y\) decomposes into a spectrum of \(\tau\) elementary mappings.
If \(X\) and \(Y\) are compacta, we obtain:
Theorem 5. An irreducible mapping of a compactum \(X\) onto a compactum \(Y\) is the product of a countable number of elementary mappings.
Corollary of Theorem 5. If \(X\to Y\) is an irreducible mapping of a compactum \(X\) onto a compactum \(Y\), then in \(Y\) there is everywhere a dense set of type \(G_\delta\) of points of one-to-oneness.
§ 3. Decomposition of an irreducible mapping of a bicompactum into a transfinite spectrum of elementary mappings. Let \(f:X\to Y\) be an irreducible mapping of a bicompactum \(X\) onto a bicompactum \(Y\). Analogously to how this was done in \((^2)\), choose in \(Y\) an open everywhere dense set \(U\) so that \(Y_1=d_XU\) differs from \(Y\). In \(Y\) choose an everywhere dense open set \(U_1\) and construct \(d_XU=Y_2\), and so on, by transfinite induction. If \(\alpha\) is a limit ordinal, then as \(Y_\alpha\) we take the limit of the spectrum of all \(Y_\beta\) for ordinals \(\beta<\alpha\).
The transfinite process of constructing the spaces \(Y_\alpha\) necessarily terminates at some cardinal number \(\Omega(X,Y)\), since the weight of the bicompactum \(X\) does not exceed \(2^\tau\), where \(\tau\) is the weight of the bicompactum \(Y\). Obviously, the limit of the spectrum of the spaces \(\{Y_\alpha\}_{\alpha\leq\Omega}\) is an irreducible image of the bicompactum \(X\). It coincides with \(X\), since otherwise one could find an open everywhere dense set \(\widetilde U\subset \widetilde Y\) with the property \(d_X\widetilde U\ne \widetilde Y\). This contradicts the fact that the process terminates at the space \(\widetilde Y\).
We have proved the following proposition:
Theorem 6. Every irreducible mapping of a bicompactum \(X\) onto a bicompactum \(Y\) decomposes into a transfinite spectrum of mappings \(\pi_{\alpha_2}^{\alpha_1}\), in which every mapping \(\pi_{\alpha}^{\alpha+1}\) is elementary.
§ 4. Incomplete absolutes. Let \(X\) be a bicompactum and let \(\eta\) be some system of its closed nowhere dense subsets, possessing the following properties: 1) if \(F_1\in\eta,\ F_2\in\eta\), then \(F_1\cup F_2\in\eta\); 2) if \(F_1\subset F_2,\ F_2\in\eta\), then \(F_1\in\eta\). We shall call the space \(X\), together with the system \(\eta\) specified in it, a pair.
Let \(f:Z\to X\) be an irreducible mapping, and suppose that for any two open sets \(U_1\) and \(U_2\) in \(Z\) whose closures are disjoint, one has
\[ f[U_1]\cap f[U_2]\in\eta . \]
We shall say that the mapping \(f\) is generated by the system \(\eta\). If \(F\in\eta\), then \(f^{-1}(F)\) is nowhere dense in \(Z\). The mapping \(f\) and the system \(\eta\) generate in \(Z\) the system
\[ \zeta=f^{-1}(\eta)=\{f^{-1}(F)\setminus F\in\eta\}. \]
Suppose that, along with the mapping \(f\), there exists a mapping \(g:Z\to Y\), and moreover
\(g^{-1}(g[U_1]\cap g[U_2])\in\zeta\) \(([U_1]\cap[U_2]=\Lambda)\).
The mapping \(g\) generates in the space \(Y\) the system \(\vartheta=\{g(F),\ F\in\zeta\}\). We shall say that the pair \((X,\eta)\) is coabsolute with the pair \((Y,\vartheta)\). It is easy to see that the mapping \(g\) is generated by the system \(\vartheta\).
Assertion. The relation of coabsoluteness is an equivalence relation.
Reflexivity and symmetry are obvious. Let us verify transitivity. Suppose \(X(\eta)\sim Y(\zeta)\), \(Y(\zeta\sim Z(\vartheta))\). This means that there exist four mappings
\[ P\to X,\quad Q\to Z,\quad P\to Y,\quad Q\to Y. \]
By Theorem 4, \(P\) and \(Q\) are limits of the spectrum of the spaces \(b(Y\setminus F)\), \(F\in\eta\). To each \(F\) there correspond two bicompact extensions \(b_p(Y\setminus F)\) and \(b_q(Y\setminus F)\).
Taking the extension \(\beta(Y\setminus F)\) for each \(F\) and constructing a spectrum from them, we obtain in the limit a space \(R\), which establishes a connection between \(X\) and \(Z\) by means of the natural mappings \(R\to P\to X,\ R\to Q\to Z\).
Thus, with the aid of the pair \((X,\eta)\) we construct a whole class of pairs \(\{(Y,\zeta)\}\), and this class \(K\) is generated by any one of its pairs \((Y,\zeta)\).
Let \(Z,Y\in K\), and suppose there exists an irreducible mapping \(f:Z\to Y\). Among all irreducible mappings \(Z\to Y\) we choose those that are generated by the system \(\zeta\). From all the selected mappings we construct the system \(\alpha\).
Considering together the spaces from the class \(K\) and all mappings from \(\alpha\), we obtain the category \(\overline K\). Let \(X\) be an arbitrary space of this category. Put \(\widehat X(\eta)=\lim\{\beta U\}\), \(u=X\setminus F,\ F\in\eta\).
Lemma 1. If \(f: Z\to X\) is the mapping generated by the system \(\eta\), then \(\widehat Z(f^{-1}(\eta))=\widehat X(\eta)\), where \(f^{-1}(\eta)\) is the system of all complete inverse images of sets from the system \(\eta\).
Lemma 2. If \(X_1\) and \(X_2\) are spaces of the category \(\overline K\), then \(\widehat X_1=\widehat X_2\).
With the aid of these lemmas it is easy to prove
Theorem 7. \(\widehat X\) is a projective object in the category \(\overline K\).
By passing to bicompact extensions, all the results obtained can be transferred to arbitrary completely regular spaces.
The author expresses his gratitude to V. I. Ponomarev for his attention and valuable advice.
Mechanics and Mathematics Faculty
Moscow State University
named after M. V. Lomonosov
Received
8 I 1969
References
\(^{1}\) N. S. Lashnev, Vestn. Mosk. Univ., Ser. Math., no. 5 (1968). \(^{2}\) V. I. Ponomarev, DAN, 149, No. 1, 26 (1963).