Last-modified: 2010-08-24 (火) 17:57:44
Every β-open cover has a finite subcover.
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.
For definition 1,
- 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.