Hattori spaces and almost topological groups

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.