We generalize the Bolzano-Weierstrass theorem (that every bounded sequence of reals admits a convergent subsequence) on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals and quotient Boolean algebras. In particular we show that an ideal does not have BW property if and only if its quotient Boolean algebra has a countably splitting family.

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We introduce the notion of equi-ideal convergence and use it to prove an ideal variant of Egorov's theorem. We also show that this variant usually cannot be strengthen to a direct analogue of Egorov's theorem.

We consider the Katětov order between ideals of subsets of natural numbers ("≤K ") and its stronger variant-containing an isomorphic ideal (" "). In particular, we are interested in ideals I for which I ≤K J ⇒ I J for every ideal J . We find examples of ideals with this property and show how this property can be used to reformulate some problems known from the literature in terms of the Katětov order instead of the order " " (and vice versa).