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β-compact の変更点


 *Definition [#e2d02c56]
 Every [[β-open]] cover has a finite subcover.
 
 *Definition 2 [#l95468f7]
 Let X be a topological space and let U be an open cover. Then X is β-compact iff every subset family N of X with the properties:
 +for A in U, we can pick R in N such that A or the complement of A is contained in R
 +the U-interiors of members of N cover X
 has a finite subfamily which covers X.
 
 *Property [#eb446f32]
 For definition 1,
 -[Ganster1987] Since [[preopen]] sets and [[semiopen]] sets are clearly [[β-open]], every β-compact space has to be [[strongly compact]] and [[semicompact]].
 -[Ganster1987] Infinite β-compact spaces do not exist.
 
 *Reference [#oea37d0d]
 :Definition 1|M.E.Abd El-Monsef, R.A.Mahmoud, A.A.Nasef and A.M.Kozae, ''Some generalized forms of compactness and closedness'', Delta J. Sci. 9(2) (1985), 257-269.
 :Definition 2| D.V.Thampuran, ''Nets and Compactness'', Portugaliae Mathematica Vol.28(1) pp.37-54.
 -M. Ganster, ''Some remarks on strongly compact spaces and semi compact spaces'', Bull. Malaysian Math. Soc. (10) 2 (1987), 67?81.