If 𝒜 is a class of open covers of
a topological space (X,𝒯 ), then (X,𝒯 ) is said to be strongly 𝒜-stable provided
that for each 𝒞∈𝒜 there is a homeomorphism h mapping X onto X, other
than the identity homeomorphism, such that for each C ∈𝒞, h(C) = C.
This paper studies strongly 𝒜-stable spaces. Although there are compact
connected metric spaces that are not even strongly 𝒜-stable with respect to the
class 𝒜 of all finite open covers, there is an extremely weak homogeneity
condition that guarantees that a space (X,𝒯 ) is strongly 𝒜-stable with
respect to the class .4 of all locally finite open covers. If H(X) is the full
homeomorphism group of a space (X,𝒯 ) that is strongly 𝒜-stable with respect to the
class 𝒜 of all finite open covers, then H(X) is nonabelian and there is a
nondiscrete Hausdorff topology 𝒯 for H(X) such that (H(X),𝒯 ) is a topological
group.
