关于Lasnev空间的超空间(英文).pdfVIP

T h is is a con trad ict ion. H ence V n (x ) is a neighbou rhood of x. T hu s X is a first coun tab le space. Since, a quasi2base is a K 2netw o rk, from T heo rem A (iii) , w e know tha t X is a L a snev space, so X is m etrizab le. Corrllary 2. 4　If X has a Ρ2CF quasi2base, then the f ollow ing sta tem en ts a re equ iva len t: ( i)X is m etriz able. ( ii)X is a L a snev sp ace. ( iii)X is a F r’échet sp ace. ( iv )X is a k′2sp ace. Remark　 It is know n tha t if a k2space has a Ρ2CF base, then it is m etrizab le. So , a quest ion m ay be ra ised: If a k2space has a Ρ2CF quasi2base, is it m etrizab le? T he quest ion is in terest ing since w e have a lready know n the rela t ion: k′+ (Ρ2CF quasi2base) α] k+ (Ρ2CF base). How ever, is th is ba lance rela t ion essen t ia l? References [1 ] J. Ceder, S om e cha racteriz a tions of m etric sp aces, Pacif ic J. M ath. , 11 (1961) , 105- 125. [2 ] R. Engelk ing, Genera l top ology , W arszaw a, 1977. [ 3 ] Ge Y ing, O n the p roblem of T. M iz okam i, Q A in Genera l Topo logy, V o l. 10 (1992) , 79- 80. [ 4 ] L iu Chuan, S p ace w ith a Ρ2com p act f in ite K2netw ork , Q A in Genera l Topo logy, V o l. 10 (1992) , 81- 87. [ 5 ] E. M ichael, T op olog ies in sp aces of subsets, T ran s Am er. M ath. Soc. , 71 (1951) , 152- 182. [6 ] T. M izokam i, O n CF f am ilies and hyp ersp aces of com p act subsets, Topo logy and its A pp lica2 t ion s, 35 (1990) , 75- 92. [ 7 ] T. M izokam i and K. Ko iw a, O n hyp ersp aces of com p act and f in ite subsets, Bu ll. Joetsu U n i2 versity of Educa t ion, 6 (1971) , 1- 14. [8 ] A. O kuyam a, N ote on hyp ersp aces of com p act subsets, M ath. Japan, 24 (1979) , 301- 305. 关于La snev 空间的超空间 谢　琳 (辽宁师范大学数学系, 大连 116029) 刘　勇 (大连铁道学院, 大连 116022) 摘 要 讨论了L a snev 空间的超空间的某些性质. 文中构造一反例, 证明存在可数L a snev 空间, 其紧 子集超空间不是层型空间. 并指出文[6 ]中关于上述结果的证明中有一关键性失误, 故[6 ]中的反例 尚不能说明上述结论成立. 本文还对具有 Ρ2CF 拟基的 k′空间给出一个刻画定理. —202— ? 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. Jou rnal o