A topological space
Y
,
τ
is called epi-
α
-normal (epi-
β
-normal) if there is a coarser topology
τ
′
on
Y
such that
Y
,
τ
′
is
T
1
α
-normal (
T
1
β
-normal). We investigate these properties and show some examples to explain the relationships of epi-
α
-normal (epi-
β
-normal) with other weaker versions of normality and some topological spaces.