IDEAL-VERSIONS OF BOLZANO-WEIERSTRASS PROPERTY

Abstract

 Let I, J be ideals on ω, we say that a space X has (I,J)-BW property if for every sequence <xn : n ∈ ω>⊂ X, there exists A∈I^+ such that the subsequence <xn : n ∈ A>is J-convergent. This is a variation of BolzanoWeierstrass property. By modifying some notions (replace conv by conv(I), I-splitting family by (I,J)-splitting family and Ramsey^* by (I,J)-Ramsey^*), we obtain many characterizations of (I,J)-BW property. As applications, we show that an ultrafilter U is a conv(I)-ultrafilter if, and only if [0,1] has (U^*,I)-BW property.