ν-compact
Last-modified: 2010-12-19 (日) 19:10:55
Definition
- 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
- 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 compact→almost 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
- Any ν-closed subset of a ν-compact space is ν-compact.
- ν-irresolute image of a ν-compact space is ν-compact.
- A space X is ν-compact if and oonly if every ν-closed subset is ν-compact.
- 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:
- X is ν-compact.
- 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:
- X is ν-compact.
- 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
- 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