Paracompactness, metacompactness, and semi-open covers
Paracompactness, metacompactness, and semi-open covers
复制标题
准紧性、亚紧性和半开盖
DOI：
10.1090/s0002-9939-1979-0516472-7
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发表时间：
1979
期刊：
影响因子：
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通讯作者：
H. Junnila
中科院分区：
文献类型：
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作者：
H. Junnila
仿紧性和亚紧性通过开覆盖的局部有限和点有限半开加细来刻画。从其中一个刻画可以得出仿紧空间在伪开紧映射下的连续象是亚紧的。1.半开放式封面。关于本文中没有定义的概念的含义，请参见[3];然而，请注意，我们不要求仿紧空间或亚紧空间满足任何分离公理。在下文中，X表示拓扑空间。设f5是X的一个覆盖。对于每个x E X，我们令（I）x = {L E I x E L}。注意，对于每个x ∈ X，我们有St（x，E）= U（fC）x。如果集合St（x，f5）是x的一个邻域，对于每个x ∈ X，那么我们说f5是X的一个半开覆盖。关于半开覆盖的一些性质，见[6]。当θ L是X的一个覆盖时，我们说θ L是覆盖f5的一个F-加细，如果每一个集合N ∈ at包含在族C的集合的某个有限并中。LEMMA 1.1.拓扑空间的局部有限半开覆盖有局部有限闭F-加细。证据设C是X的局部有限半开覆盖。对于CS的每个子族C'，令K（C'）= Cl（n C '）Int（U（C '））。注意，如果CD'是无限的，则K（C'）=0。对于每个C' c CS，我们有K（E'）C U C '。要看到这一点，令x E K（E '）。然后x 4 Int（U（CS '）），并且由于x E int（U（C）j），因此我们有（Ox n Ct' #0，换句话说，x E U C '。由于对于每个x ∈ X都有x ∈ K（（fE）x），因此从前面可以得出闭族Yu = {K（fE ′）I C ′ c C5）是C5的F-加细。为了证明'XC是局部有限的，设x ∈ X。由于C是局部有限的，所以子族C5* = { L E Cix E L）是有限的，并且开集0 = XCl（U（f-E *））包含x。如果C' c和K（'）n 0 #0，则[cl（n f）] n 0 0，因此（n t）n 0 #0。因此，如果K（C '）n 0 ≤ 0，则CS' c E 5 *;因此x的邻域0只与族YuC的1/2个集合相交。O 1978年1月4日以“On pseudo-open mappings of paracompact spaces”为题提交给学会;编辑于1978年3月3日收到，并于1978年6月19日以修订形式收到。AMS（MOS）主题分类（1970年）。第54 D20集
Paracompactness and metacompactness are characterized in terms of locally finite and point-finite semi-open refinements of open covers. It follows from one of these characterizations that a continuous image of a paracompact space under a pseudo-open and compact mapping is metacompact. 1. On semi-open covers. For the meaning of concepts used without definition in this paper, see [3]; note, however, that we do not require paracompact spaces or metacompact spaces to satisfy any separation axioms. Throughout the following, X denotes a topological space. Let f5 be a cover of X. For each x E X, we let (I) x = {L E I x E L}. Note that we have St(x, E) = U (fC)x for each x E X. If the set St(x, f5) is a neighborhood of x for each x E X, then we say that f5 is a semi-open cover of X. For some properties of semi-open covers, see [6]. When 9L is a cover of X, we say that 9, is an F-refinement of the cover f5 if each set N E at is contained in some finite union of sets of the family C. LEMMA 1.1. A locally finite semi-open cover of a topological space has a locally finite closed F-refinement. PROOF. Let C be a locally finite and semi-open cover of X. For each subfamily C' of CS, let K(C') = Cl(n C') Int(U (C ')). Note that if CD' is infinite, then K(C') =0. For each C' c CS, we have K(E') C U C'. To see this, let x E K(E'). Then x 4 Int(U (CS ')) and it follows, since x E int( U (C)j), that we have (Ox n Ct' #0, in other words, x E U C '. Since x E K((fE)x) for each x E X, it follows from the foregoing that the closed family Yu = {K(fE')I C' c C5) is an F-refinement of C5. To show that 'XC is locally finite, let x E X. Since C is locally finite, the subfamily C5* = { L E CIx E L) is finite and the open set 0 = XCl(U(f -E*)) contains x. If C' c and K(') n 0 #0, then [cl(n f )] n o 0 and hence (n t) n 0 #0. It follows that if K(C') n 0 #0, then CS' c E5*; hence the neighborhood 0 of x intersects only finitely many sets of the family YuC. O Presented to the Society, January 4, 1978 under the' title On pseudo-open mappings of paracompact spaces; received by the editors March 3, 1978 and, in revised form, June 19, 1978. AMS (MOS) subject classifications (1970). Primary 54D20.