楊春梅,李祖泉

(杭州師范大學理學院,浙江 杭州 310036)

函數(shù)空間Ck(X)上的T-Tightness和Set-Tightness

楊春梅,李祖泉*

(杭州師范大學理學院,浙江 杭州 310036)

討論了函數(shù)空間Ck(X)在賦予緊開拓撲下的T-tightness和set-tightness性質(zhì),利用開k覆蓋獲得了Ck(X)是T-tightness空間和set-tightness空間的兩個對偶定理,將點態(tài)收斂拓撲函數(shù)空間Cp(X)的相關結論推廣到緊開拓撲函數(shù)空間Ck(X)上.

函數(shù)空間;緊開拓撲;T-tightness;set-tightness;k覆蓋

0 引 言

T-tightness空間和set-tightness空間是弱于tightness空間從而弱于第一可數(shù)空間的一類拓撲空間.set-tightness空間是由Arhangel’skii A V[1]引入,最初稱為擬特征(quasi-character)空間,后來Juhasz I[2]將其稱為set-tightness空間,Bella A[3-5]對其進行了系統(tǒng)研究.Juhasz I為了比較set-tightness空間和tightness空間引入T-tightness空間.函數(shù)空間Ck(X)的各種類型tightness特征中已經(jīng)有tightness, fan tightness和可數(shù)強fan tightness對偶性的證明,上述性質(zhì)已推廣到集值映射空間Ck(X)[6-7]上.函數(shù)空間Cp(X)中的T-tightness和set-tightness性質(zhì)的刻畫是由Sakai M[8]得到的.Ck(X)空間的拓撲性質(zhì)與Cp(X)有很大區(qū)別.McCoy R A, Ntantu I[9]和林壽等[10]對Ck(X)的拓撲性質(zhì)均有系統(tǒng)的論述.在此給出了具有緊開拓撲函數(shù)空間Ck(X)的T-tightness和set-tightness性質(zhì)與基本空間X的對偶定理,獲得了Ck(X)是T-tightness和set-tightness空間的等價性證明.

1 預備知識

在此拓撲空間X是Tychonoff的,λ,τ和κ表示無限基數(shù),ω表示可數(shù)序數(shù)及基數(shù),cf(κ)表示最小的基數(shù)λ使得κ具有一個基數(shù)為λ的共尾子集,即cf(κ)=min{λ:|A|=λ<κ,A與κ共尾}.如果κ≥ω,并且cf(κ)=κ,則稱κ是正則基數(shù).若κ是正則基數(shù),A?κ,|A|<κ,則supA<κ.對于κ≥ω,cf(κ)是正則基數(shù).R表示實直線,Ck(X)為X到R上的所有的連續(xù)映射族,并且賦予緊開拓撲.文中未定義的術語和符號均以[10-11]為準.

函數(shù)空間Ck(X)上的子基開集形如[K;U]={f∈Ck(X):f(K)?U},其中K是X的非空緊子集,U是R中非空開集.

Ck(X)上的基開集形如 [K1,K2,…,Kn;U1,U2,…,Un]={f∈Ck(X):f(Ki)?Ui,1≤i≤n},

其中Ki(1≤i≤n)是X的非空緊子集,Ui(1≤i≤n)是R中非空開集.

2 函數(shù)空間Ck(X)上的T-Tightness

定義1空間X的T-tightness[2]定義為

T(X)=ω+min{τ:cf(κ)>τ,{Fα:α<κ}是X的遞增閉集族,那么∪{Fα:α<κ}是X中閉集}.

為了刻畫Ck(X)上的T-tightness性質(zhì),類似于[8]的T(τ)定義,給出下面的Tk(τ)定義.

定理1對于空間X,下列結論是等價的:

1)T(Ck(X))≤τ; 2)X具有性質(zhì)Tk(τ).

3 函數(shù)空間Ck(X)上的Set-Tightness

類似于[8]的ts-對和ts(τ)的定義,給出下面的tk-對和tk(τ)定義.

定理2對于空間X,下列結論是等價的:

1)ts(Ck(X))≤τ; 2)X具有性質(zhì)tk(τ).

[1] Arhangel’skii A V, Isler R, Tironi G. On pesudo radial spaces[J]. Comment Math Univ Caroin,1986,27:137-154.

[2] Juhasz I. Variations on tightness[J]. Studia Sci Math Hungar,1989,24:179-186.

[3] Bella A. On set tightness and T-tightness[J]. Comment Math Univ Caroin,1986,27:805-814.

[4] Bella A. Free sequences in pseudo radial spaces[J]. Comment Math Univ Caroin,1986,27:167-170.

[5] Bella A. A couple of questions concerning cardinal invariants[J]. Questions Answers Gen Topology,1996,14:139-143.

[6] 李祖泉.集值映射空間上的Tightness和Fan Tightness[J].數(shù)學研究與評論,2008,28(4):1007-1012.

[7] 郭先一,李祖泉.集值映射空間上可數(shù)強Fan Tightness[J].杭州師范大學學報:自然科學版,2010,9(1):23-25.

[8] Sakai M. Variations on tightness in function spaces[J]. Topology Appl,2000,101:273-280.

[9] McCoy R A, Ntantu I. Topological properties of spaces of continuous functions[C]//Lecture Notes in Math.No 1315. Berlin: Springer-Verlag,1988:62-66.

[10] 林壽.度量空間與函數(shù)空間的拓撲[M].北京:科學出版社,2004:204-213.

[11] Engelking R. General Topology[M]. Warszwa: Polish Scientific Publishers,1977:87-287.

T-TightnessandSet-TightnessinFunctionSpaceCk(X)

YANG Chun-mei, LI Zu-quan

(College of Science, Hangzhou Normal University, Hangzhou 310036, China)

This paper discussed the T-tightness and set-tightness properties in function spaceCk(X) with compact-open topology obtained two dualities ofCk(X) being T-tightness space and set-tightness space by means of openingk-cover and generalized the related results from point-wise convergence topology spaceCp(X) to compact-open topology spaceCk(X).

function space; compact-open topology; T-tightness; set-tightness;k-cover

10.3969/j.issn.1674-232X.2011.02.007

2010-09-20

楊春梅(1986—),女,吉林松原人,基礎數(shù)學專業(yè)碩士研究生,主要從事一般拓撲學研究.

*通信作者：李祖泉(1963—)，男,吉林懷德人,教授,主要從事一般拓撲學研究.E-mail: hzsdlzq@sina.com

O189.1MSC201054C35

A

1674-232X(2011)02-0124-03