β-compact

Definition

Every β-open cover has a finite subcover.

Definition 2

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:

1. for A in U, we can pick R in N such that A or the complement of A is contained in R
2. the U-interiors of members of N cover X
   has a finite subfamily which covers X.

Property

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

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.