countably ν-compact

Definition

• Topological space X is said to be countably ν-compact space if every countable ν-open cover of it has a finite sub cover.

Property

• ν-closed subset of a countably ν-compact space is countably ν-compact.
• A ν-irresolute image of a countably ν-compact space is countably ν-compact.
• countable product of countably ν-compact spaces is countably ν-compact.
• countable union of countably ν-compact spaces is countably ν-compact.
• For a ν-T_1 topological space X the following statements are equivalent
  1. X is countably ν-compact.
  2. Every countable family of ν-closed subsets of X which has the finite intersection property has a non-empty intersection.
  3. Every infinite subset has an ν-accumulation point.
  4. Every sequence in X has a ν-limit point.
  5. Every infinite ν-open cover has a proper sub cover

Reference

• S. Balasubramanian, P. Aruna Swathi Vyjayanthi and C. Sandhya,ν-Compact spaces , Scientia Magna, international book series, Vol. 5 (2009), No. 1 (78-82)