For a subset A of an almost topological group, the Hattori space H (A) is a topological space whose underlying set is G and whose topology is defined as follows: If x ∈ A (respectively, x ∉ A), then the neighborhoods of x in H (A) are the same neighborhoods of x in the reflection group (respectively, G). In this paper, we show that: i) The family of topologies H (G) on a proper almost topological group G forms a complete lattice. ii) If G is an almost topological group such that e (G) ≤ κ where κ = ℵ 0 or κ is uncountable with cof (κ) > ℵ 0, then for every A ⊆ G and f: H (A) → R continuous function, the set B = x ∈ G | f is discontinuous at x in G ⁎ has cardinality at most κ. iii) Under OCA the following are equivalent for A ⊆ R: a) R ∖ A is countable. b) C p (H (A) ) is normal. c) C p (H (A) ) is Lindelöf. Answering Problem 7. 7 from 7 iv) We provide a method for constructing examples of almost topological groups. v) We can concluded that every almost topological group admits an embedding into a saturated group which is not itself an almost topological group.
Calderón-Villalobos et al. (Sun,) studied this question.