Abstract
Full Text
MATHEMATICS
V. I. PONOMAREV
ON \((\omega, p)\)-MAPPINGS OF TOPOLOGICAL SPACES
(Presented by Academician P. S. Aleksandrov on 23 XII 1964)
1. Definition and the main lemma*
Basic definition. Let \(\omega\) be some fixed covering of a space \(X\). A continuous mapping \(f:X \to Y\) of the space \(X\) onto the space \(Y\) will be called an \((\omega,p)\)-mapping if, for every point \(y \in Y\), there exist a system \(\omega_y=\omega_{f^{-1}y} \subseteq \omega\) with property \(p\) in its body \(\widetilde{\omega}_y\), and a neighborhood \(V_y\) of the point \(y\) in \(Y\), such that
\[ f^{-1}V_y \subseteq \widetilde{\omega}_y \]
(from which, in particular, it follows that \(\omega_y\) covers the set \(f^{-1}y\)).**
Main lemma. Let \(f:X \to Y\) be, for a given covering \(\omega\) of the space \(X\), a continuous \((\omega,p)\)-mapping of the space \(X\) onto a paracompact (respectively, strongly paracompact) space \(Y\), where \(p\) is the property of local finiteness (respectively, countability). Then into the covering \(\omega\) of the space \(X\) one can inscribe a locally finite (respectively, star-countable) covering \(\omega'\).
Proof. For each point \(y \in Y\), choose a subsystem \(\omega_y \subseteq \omega\) with property \(p\) (i.e., a subsystem locally finite in the body \(\widetilde{\omega}_y\), respectively countable) covering the set \(f^{-1}y\), and a neighborhood \(V_y\) of the point \(y \in Y\) in such a way that
\[ f^{-1}V_y \subseteq \omega_y . \]
Then \(\Omega=\{V_y\}\) is an open covering of the space \(Y\) (here \(y\) runs through all points of the space \(Y\)). Into the covering \(\Omega\) we inscribe a locally finite covering \(\Omega'=\{V_\alpha'\}\), if the space \(Y\) is paracompact, and a star-finite one if \(Y\) is strongly paracompact. Now, for each element \(V_\alpha' \in \Omega'\), take some set \(V_{y_\alpha} \in \Omega\) containing the set \(V_\alpha'\). To this \(V_{y_\alpha}\) there corresponds a subsystem \(\omega_{y_\alpha} \subseteq \omega\) with property \(p\), such that \(f^{-1}V_{y_\alpha} \subseteq \widetilde{\omega}_{y_\alpha}\). Now denote by \(\omega_\alpha'\) the system of open sets in \(X\) obtained by intersecting the elements of the system \(\omega_{y_\alpha}\) with the open set \(f^{-1}V_\alpha'\) in \(X\). Consider the system
\[ \omega'=\bigcup_\alpha \omega_\alpha' . \]
This system \(\omega'\) is inscribed in the covering \(\omega\) and is a locally finite covering of the space \(X\), if \(p\) is the property of local finiteness (i.e., all \(\omega_y\) are locally finite in \(\widetilde{\omega}_y\)), \(Y\) is paracompact, and is a star-countable covering of the space \(X\), if \(p\) is the property of countability and \(Y\) is strongly paracompact. The lemma is proved.***
* In this paper all spaces are assumed to be regular, and all mappings continuous.
** The body \(\widetilde{\omega}_A\) of the system \(\omega_A\) is, as usual, the union of the elements contained in \(\omega_A\). The system \(\omega_A\) is locally finite in the body \(\widetilde{\omega}_A\) (in the whole space \(X\)) if, for every point \(x \in \widetilde{\omega}_A\) (for every point \(x \in X\)), there exists a neighborhood \(Ux \subseteq X\) that intersects only finitely many elements of the system \(\omega_A\). The letter \(p\) may mean the property of local finiteness of \(\omega_A\) in \(\widetilde{\omega}_A\), star-finiteness, or simply countability or finiteness.
*** We also note that \(\omega'\) will be a star-finite covering of the space \(X\), if \(p\) is the property of star-finiteness and the space \(Y\) is strongly paracompact.
2. Some consequences of the main lemma.
Proposition 1. In order that a space \(X\) be paracompact, it is sufficient that for every open covering \(\omega\) of this space there exist a continuous \((\omega,p)\)-mapping \(f\) of the space \(X\) onto a paracompact \(Y_\omega\). The property \(p\) denotes local finiteness.
Remark. It is not hard to verify that a one-to-one perfect mapping \(f\) of a space \(X\) onto a space \(Y\) will always be an \((\omega,p)\)-mapping for every open covering \(\omega\) of the space \(X\), and here \(p\) will even be the property of finiteness. From this remark we obtain the known theorem (see, for example, \((^5)\)): if \(f\) is a one-to-one perfect mapping of a space \(X\) onto a paracompact \(Y\), then the space \(X\) is also paracompact.
Proposition 2. Suppose that in the space \(X\) there is a countable refining system \(\mathfrak A=\{\omega_i\}\) of open coverings. Suppose further that for each \(i\) there exists a continuous \((\omega_i,p)\)-mapping of this space onto some paracompact \(Y_i\), where \(p\) is the property of local finiteness. Then the space \(X\) is necessarily metrizable.
Proposition 3. In order that a space \(X\) be strongly paracompact, it is sufficient that for every open covering \(\omega\) of this space there exist a continuous \((\omega,p)\)-mapping \(f\) of the space \(X\) onto a strongly paracompact space \(Y_\omega\). Here \(p\) may denote either the property of countability or the property of star-finiteness.
Let us note that a closed \(S\)-mapping** \(f\) of a space \(X\) onto a space \(Y\) will always be an \((\omega,p)\)-mapping for any open covering \(\omega\) of the space \(X\), where \(p\) is the property of countability. Therefore from our Proposition 3 it follows that
Proposition 4. If \(f\) is a closed \(S\)-mapping of a space \(X\) onto a strongly paracompact space \(Y\), then the space \(X\) is also strongly paracompact.
3. Fully paracompact spaces.
A space \(X\) is called fully paracompact if into every one of its open coverings \(\omega\) one can weakly inscribe a \(\sigma\)-star-finite covering \(\omega'\). Fully paracompact spaces are necessarily paracompact; at the same time there exist fully paracompact spaces that are not strongly paracompact (see \((^4)\)). Fully paracompact metrizable spaces are also called strongly metrizable*—these are spaces in which there exists a \(\sigma\)-star-finite base.
Proposition 5. In order that a normal space \(X\) be fully paracompact, it is sufficient (and, obviously, necessary) that into every one of its open coverings \(\omega\) one can weakly inscribe a \(\sigma\)-star-countable covering \(\omega'\).
From this proposition and from the main lemma it follows:
Proposition 6. Each of the following properties of a space \(X\) is sufficient (and necessary) for its strong metrizability:
A. The space \(X\) has a \(\sigma\)-star-countable base.
B. The space \(X\) is metrizable and in it there is such a base \(\mathfrak B=\bigcup_{i=1}^{\infty}\omega_i\),
* A countable system \(\mathfrak A=\{\omega_i\}\) of open coverings of a space \(X\) is refining if for any point \(x\) of this space and any neighborhood \(Ux\) of it there is a covering \(\omega_i\in\mathfrak A\) such that the sum \(\Gamma_i x\) of its elements containing the point \(x\) is contained in \(Ux\) (in connection with this concept see the papers \((^2,^3,^8)\)).
* A continuous mapping \(f:X\to Y\) is called an \(S\)-mapping* if the inverse image of every point \(y\in Y\) is a finally compact space (see the papers \((^{6-8})\)).
* A covering \(\omega\) is weakly inscribed** in a covering \(\omega\) of a space \(X\) if there exists a subcovering \(\omega''\) of the covering \(\omega'\) inscribed in \(\omega\).
* A system \(\mathfrak B\) of subsets of a space \(X\) is called *\(\sigma\)-star-finite (\(\sigma\)-star-countable or \(\sigma\)-locally finite)) if the system is the union of a countable number of star-finite (star-countable or locally finite) coverings of this space \(X\).
where each \(\omega_i\) is an open cover, such that for each \(i\) there exists an \((\omega_i,p)\)-mapping\(^*\) \(f_i:X\to Y_i\) onto a strongly paracompact \(Y_i\).
B. In the space \(X\) there is a countable refining system \(\mathfrak A=\{\omega_i\}\) of open covers, and for each \(i\) there exists an \((\omega_i,p)\)-mapping of the space \(X\) onto a strongly paracompact \(Y_i\).
4. Mappings into Baire space\(^ {**}\). The following arguments are adjacent to the work of Yu. M. Smirnov \((^6)\).
A system of sets \(\eta=\{U_\lambda\}\) of a star-finite cover \(\omega\) is called connected (see \((^{1,6})\)) if for every pair \(U,U'\) of sets from \(\eta\) there exist sets \(U_1,\ldots,U_k\) from \(\eta\) such that \(U_i\cap U_{i+1}\ne \Lambda\) for \(i=1,2,\ldots,k-1\), and \(U_1\cap U\ne \Lambda,\ U_k\cap U'\ne \Lambda\). A maximal connected subsystem \(\eta\) of the system \(\omega\) is called a component of the system \(\omega\). It should be noted that every component of a star-finite system \(\omega\) is always at most countable, and the body of every component, i.e. the sum of the elements it contains, is an open-and-closed set of the space \(X\) (see \((^{1,6})\)). Let the components of the star-finite cover \(\omega\) be \(\eta_\alpha\), where \(\alpha\) are distinct indices, and let \(\widetilde{\eta}_\alpha\) be the bodies of these components. By \(D_\omega\) denote the set of these distinct indices, endowed with the discrete topology. Obviously, \(D_\omega\) is topologically contained in the Baire space \(B^\tau\), where \(\tau\) is the cardinality of the set \(D_\omega\). Let, further, \(x\in X\) be an arbitrary point. Since the system \(\Omega\) of all \(\widetilde{\eta}_\alpha\) is a disjoint open-and-closed cover of the space \(X\), there is a unique set \(\widetilde{\eta}_\alpha\) containing the point \(x\). We obtain a mapping \(f_\omega\) of the space \(X\) onto the discrete space \(D_\omega=\{\alpha\}\): \(f_\omega x=\alpha\), where \(\alpha\) is the index of the set \(\widetilde{\eta}_\alpha\ni x\). This mapping is continuous and, moreover, is an \((\omega,p)\)-mapping, where \(p\) is the property of countability. Thus, we have proved
Lemma 1. If \(\omega\) is a star-finite cover of the space \(X\), then there exists a continuous \((\omega,p)\)-mapping of this space \(X\) into Baire space.
Let, further, the space \(X\) have a countable set \(\mathfrak A\) of star-finite covers \(\omega_i\). We proceed as follows: in each cover \(\omega_i\) we consider its components \(\eta_{\alpha i}\), the bodies \(\widetilde{\eta}_{\alpha i}\) of these components (which are open-and-closed sets of the space \(X\)), and also the open-and-closed covers \(\Omega_i\) of the space \(X\), consisting of the open-and-closed sets \(\widetilde{\eta}_{\alpha i}\) for fixed \(i\). Let now \(x\) be an arbitrary point of \(X\). For each \(i\) this point belongs to the body \(\widetilde{\eta}_{\alpha i}\) of some, and moreover unique, component \(\eta_{\alpha i}\) of the cover \(\omega_i\), i.e. to a unique element of the open-and-closed cover \(\Omega_i\). To this point \(x\) we put in correspondence the countable sequence \(\{\alpha i\}\) of indices, i.e. the point of the space
\[ B^\tau=\prod_{i=1}^{\infty}D_{\omega_i}, \]
where \(\tau\) is the greatest cardinality of the sets \(D_{\omega_i}\). In this way we have obtained a mapping \(f\) of the space \(X\) into the space \(B^\tau\). This mapping is continuous and, moreover, is an \((\omega_i,p)\)-mapping for every \(i\) (\(p\) is the property of countability). In addition
\(^*\) Beginning from this point, \(p\) is the property of countability.
\(^ {**}\) The space \(B^\tau\) may be defined, for example, as follows: let \(W\) be some set of indices of cardinality \(\tau\). The points of the space \(B^\tau\), by definition, are countable sequences \((\alpha_1,\ldots,\alpha_i,\ldots)\) of these indices from \(W\). If \(x=(\alpha_1,\ldots,\alpha_i,\ldots)\) and \(y=(\beta_1,\ldots,\beta_i,\ldots)\) are two points of \(B^\tau\), then we put \(\rho(x,y)=1/k(x,y)\), where \(k(x,y)\) is the first natural number \(i\) such that \(\alpha_i\ne \beta_i\). The space \(B^\tau\), thus, is a metric space of weight \(\tau\) and dimension \(\dim B^\tau=0\), containing (in the sense of topological embedding) all metrizable spaces of weight \(\tau\) and dimension \(\dim\) equal to zero. The space \(B^\tau\) may be defined also as the topological product of a countable number of discrete spaces \(D_i\) of cardinality \(\tau\). On Baire space see \((^{6,8,9})\).
therefore, every set \(\eta_{a_i}\) will be marked\(^*\) for the mapping \(f\). Thus, it has been proved:
Lemma 2. If in the space \(X\) there exists a countable set \(\mathfrak A\) of star-finite covers \(\omega_i\), then there exists a continuous mapping into Baire space \(B^\tau\) which is an \((\omega_i,p)\)-mapping for every \(i\).
From Lemmas 1 and 2 the following propositions immediately follow:
Proposition 7. For every open cover \(\omega\) of a strongly paracompact space \(X\) there exists a continuous \((\omega,p)\)-mapping of it into Baire space (\(p\) is the property of countability).
Proposition 8. Let the space \(X\) be strongly metrizable, and let \(\mathfrak B\) be its base, decomposable into the sum of a countable number of star-finite covers \(\omega_i\). Then there exists a continuous \((\omega_i,p)\)-mapping (for every \(i\)) \(f\) of this space \(X\) into Baire space \(B^\tau\); moreover, \(f\) automatically turns out to be an \(S\)-mapping.
5. On quasicomponents and quasimonotone mappings. A closed set \(A\) of a topological space \(X\) is called quasiconnected if, for every open-and-closed set \(U \subseteq X\) for which \(U \cap A \ne \Lambda\), necessarily \(A \subseteq U\). A maximal quasiconnected set \(A\) is called a quasicomponent of the space \(X\). A continuous mapping \(f : X \to Y\) is called quasimonotone if for every point \(y \in Y\) the set \(f^{-1}y\) is quasiconnected in \(X\). We note that every quasicomponent of a fully paracompact space \(X\), and every quasiconnected subspace of a fully paracompact space \(X\), is finally compact.
The following basic result holds:
Proposition 9\({}^{**}\). Let \(f : X \to Y\) be a quasimonotone quotient mapping of a strongly paracompact (respectively, fully paracompact) space \(X\) onto a space \(Y\). Then the space \(Y\) is also strongly paracompact (respectively, fully paracompact). In particular, the decomposition space \(Z\) of a strongly paracompact (respectively, fully paracompact) space \(X\) into its quasicomponents will be strongly paracompact (respectively, fully paracompact).
Remark. Every quasimonotone quotient mapping \(f\) of a space \(X\) onto a space \(Y\) will necessarily be an \((\omega,p)\)-mapping for every star-finite cover \(\omega\) of the space \(X\); if the space \(X\) is strongly paracompact, then the quotient quasimonotone mapping \(f : X \to Y\) will be an \((\omega,p)\)-mapping for every open cover \(\omega\) of the space \(X\). We note also that a quasimonotone quotient mapping \(f\) of a strongly paracompact (and also of a fully paracompact) space \(X\) onto a space \(Y\) will automatically be an \(S\)-mapping. Finally, in the case when all open-and-closed sets \(U \subseteq X\) are marked for the quotient mapping \(f : X \to Y\), the mapping \(f\) will, in addition, be quasimonotone.
Moscow State University
named after M. V. Lomonosov
Received
22 X 1964
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\(^*\) A set \(M \subseteq X\) is called marked for a mapping \(f : X \to Y\) if \(M = f^{-1}M\).
\({}^{**}\) Cf. this proposition with the main theorem in \((^{10})\). A mapping \(A : X \to Y\) is called quotient if a set \(V \subseteq Y\) is open if and only if the set \(f^{-1}V\) is open in the space \(X\).