Pseudocompact rectifiable spaces

A topological space G is said to be a rectifiable space   provided that there are a surjective homeomorphism φ:G×G→G×Gφ:G×G→G×G and an element e∈Ge∈G such that π1∘φ=π1π1∘φ=π1 and for every x∈Gx∈G we have φ(x,x)=(x,e)φ(x,x)=(x,e), where π1:G×G→Gπ1:G×G→G is the projection to the first coordinate. We firstly define the concept of rectifiable completion of rectifiable spaces and study some properties of rectifiable complete spaces, and then we mainly show that: (1) Each pseudocompact rectifiable space G is a Suslin space, which gives an affirmative answer to V.V. Uspenskijʼs question (Uspenskij, 1989 [29]); (2) Each pseudocompact infinite rectifiable space contains a non-closed countable set; (3) Each pseudocompact rectifiable space G is sequentially pseudocompact; (4) Each infinite pseudocompact rectifiable space with a continuous weak selection is homeomorphic to the Cantor set; (5) Each first-countable ω-narrow rectifiable space has a countable base. Moreover, some examples of rectifiable spaces are given and some questions concerning pseudocompactness on rectifiable spaces are posed.