Let (X, T) be a minimal transformation group with compact Hausdorff phase space. We show that if
ϕ
:
(
X
,
T
)
→
(
Y
,
T
)
\phi :(X,T) \to (Y,T)
is a distal homomorphism and has a structure similar to the structure Furstenberg derived for distal minimal sets, then for T belonging to a class of topological groups T, the homomorphism
X
→
X
/
S
(
ϕ
)
X \to X/S(\phi )
has connected fibers, where
S
(
ϕ
)
S(\phi )
is the relativized equicontinuous structure relation. The class T is defined by Sacker and Sell as consisting of all groups T with the property that there is a compact set
K
⊆
T
K \subseteq T
such that T is generated by each open neighborhood of K. They show that for such T, a distal minimal set which is a finite-to-one extension of an almost periodic minimal set is itself an almost periodic minimal set. We provide an example that shows that the restriction on T cannot be dropped. As one of the preliminaries to the above we show that given
ϕ
:
(
X
,
T
)
→
(
Y
,
T
)
\phi :(X,T) \to (Y,T)
, the relation
R
c
(
ϕ
)
Rc(\phi )
induced by the components in the fibers relative to
ϕ
\phi
, i.e.,
(
x
,
x
′
)
∈
R
c
(
ϕ
)
(x,x’) \in Rc(\phi )
if and only if x and
x
′
x’
are in the same component of
ϕ
−
1
(
ϕ
(
x
)
)
{\phi ^{ - 1}}(\phi (x))
, is a closed invariant equivalence relation. We also consider the question of when a minimal set (X, T) is such that
Q
(
x
)
Q(x)
is finite for some x in X, where Q is the regionally proximal relation. This problem was motivated by Veech’s work on almost automorphic minimal sets, i.e., the case in which
Q
(
x
)
Q(x)
is a singleton for some x in X.