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p-compact

Last-modified: 2010-09-03 (金) 00:40:51

Definition Edit

p denotes a free ultrafilter on ω, the set of natural numbers.
Let X be a topological space, (S_n) a sequence of nonempty subsets in X. A point x in X is called a p-limit point of (S_n) if for all neighborhood V of x, imgtex.fcgi?%5bres=100%5d%7b$%5c%7bn%5cin%5comega:V%5ccap%20S_n%5cneq%5cemptyset%5c%7d%5cin%20p$%7d%25.png.
If (x_n) is a sequence of points in X, "a p-limit point of (x_n)" means a p-limit point of a sequence of singleton ({x_n}).
X is said to be p-compact if every sequence (x_n) has a p-limit point.

Remark Edit

Reference Edit

  • A. R. Bernstein, A new kind of compactness for topological spaces, Fund.Math. 66 (1970), 185-193.