Consider only metrizable spaces. The notion of a slice-trivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact
U
V
∞
U{V^\infty }
point images to be slice-trivial. Theorem 4.5 posits a number of necessary and sufficient conditions for a map to be a hereditary shape equivalence. Several applications of these two theorems are made, including the following. Theorem 5.1. A cell-like map
f
:
X
→
Y
f:X \to Y
is a hereditary shape equivalence if there is a sequence
{
K
n
}
\{ {K_n}\}
of closed subsets of
Y
Y
such that (1)
Y
−
⋃
n
=
1
∞
K
n
Y - \bigcup \nolimits _{n = 1}^\infty {{K_n}}
is countable dimensional, and (2)
f
|
f
−
1
(
K
n
)
:
f
−
1
(
K
n
)
→
K
n
f|{f^{ - 1}}({K_n}):{f^{ - 1}}({K_n}) \to {K_n}
is a hereditary shape equivalence for each
n
≥
1
n \geq 1
. Theorem 5.9. If
f
:
X
→
Y
f:X \to Y
is a proper onto map whose point inverses are
U
V
∞
U{V^\infty }
sets, then
Y
Y
is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if
Y
Y
is countable dimensional, then
Y
Y
is an absolute neighborhood retract. Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.