A non-ω-balanced paratopological group with countable π-character

In this paper, we mainly prove that if G is a saturated paratopological group with we(G)≤κ, where κ is an infinite cardinal, then G is κ-narrow. It gives a partial answer to the problem posed by Sánchez in [10, Problem 2.24] and even more. Applying this property, we show that if G is a saturated Hausdorff paratopological group with we(G)πχ(G)≤ω, then G can be condensed onto a Hausdorff space with a countable base. Also, we construct a Hausdorff paratopological group with countable π-character, but it is not ω-balanced. This gives a negative answer to the problem posed by Sánchez in [9, Problem 2.13].