A continuous map
f
:
X
→
Y
f:X\to Y
of topological spaces
X
,
Y
X, Y
is said to be almost
1
1
-to-
1
1
if the set of the points
x
∈
X
x\in X
such that
f
−
1
(
f
(
x
)
)
=
{
x
}
f^{-1}(f(x))=\{x\}
is dense in
X
X
; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and
σ
\sigma
-compact spaces (e.g.,
n
n
-manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if
f
f
is a minimal self-mapping of a 2-manifold
M
M
, then point preimages under
f
f
are tree-like continua and either
M
M
is a union of 2-tori, or
M
M
is a union of Klein bottles permuted by
f
f
.