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ν-compact

Last-modified: 2010-12-19 (日) 19:10:55

Definition Edit

  • A subset A of a topological space X is said to be ν-compact if every ν-open cover has a finite sub cover.

Property Edit

  • Each ν-compact metrizable space is finite.
  • Each ν-compact and semiregular space X is compact.
  • If A ⊂ X is Almost ν-regular and compact, then A^- is ν-compact.
  • Every almost ν-regular and almost compact? subset A of the space X is ν-compact.
  • Every weak almost regular and nearly compact subset A of X is ν-compact.
  • Let A be any dense almost ν-regular subset of a space X such that every ν-open covering of A is a ν-open covering of X. Then X is almost compact if and only if X is ν-compact.
  • If a space X is weakly compact and almost regular, then X is ν-compact.
  • semi compact?→ν-compact→nearly compactalmost compact?weakly compact
  • If A is ν-compact subspace of a topological space X, the A is ν-compact relative to X.
  • Let X be any topological space. Then
    1. Any ν-closed subset of a ν-compact space is ν-compact.
    2. ν-irresolute image of a ν-compact space is ν-compact.
    3. A space X is ν-compact if and oonly if every ν-closed subset is ν-compact.
    4. A space X=ΠX_i is ν-compact if and only if every X_i is ν-compact.
  • In a space X, the following are equivalent:
    1. X is ν-compact.
    2. For every family of ν-closed sets in X satisfying empty intersection, there is finite subfamily whose intersection is empty.
  • In a space X, the following are equivalent:
    1. X is ν-compact.
    2. For every family of ν-closed sets with finite intersection property has a non empty intersection.
  • If S is an arbitrary ν-compact subset of a topological space, then every infinite subset of S has a ν-accumulation point.
  • If S is an arbitrary ν-compact subset of a topological space, then every infinite subset of S has a ω-accumulation point.
  • If f : X → Y is almost continuous, X is ν-compact and Y be a topological space then Y is ν-compact.
  • The ν-irresolute image of any ν-compact space in any Hausdorff space is ν-closed.
  • Every ν-compact, ν-Hausdorff space is almost ν-regular?.

Reference Edit

  • S. Balasubramanian, C.Sandhya and P.Aruna Swathi Vyjayanthi, Note on Regularity and ν-compactness, Int. J. Contemp. Math. Sciences, Vol. 5, (2010) no. 16, 777-784