A note on fibers and Vietoris topologies of paratopological groups

Abstract

If f: G → Y is an irreducible closed continuous mapping defined on a regular weakly collectionwise normal first-countable meta-Lindelöf locally σ paratopological group G onto a T ₁-space Y which has a neighborhood ω^ω -base at a point y ∈ Y, then f ⁻¹(y) is σ-compact in G. We prove that if a Fréchet-Urysohn space X has strong α ₄ -property and a weakly countably complete base , then X is first-countable, where M is a separable and metrizable space and = {F: F is a non-empty compact subset of M } and with the Vietoris topology. By this result we can get the first-countability of certain paratopological groups.