Let
f
:
X
′
→
X
f:X’ \to X
be a cell-like map between metric spaces and set
N
f
=
{
x
∈
X
:
f
−
1
(
x
)
≠
point}
{N_f} = \{ x \in X:{f^{ - 1}}(x) \ne {\text {point\} }}
. Even if
N
f
⊂
⋃
n
=
1
∞
B
n
{N_f} \subset \bigcup \nolimits _{n = 1}^\infty {{B_n}}
, where each
B
n
{B_n}
is closed and each
f
|
f
−
1
(
B
n
)
:
f
−
1
(
B
n
)
→
B
n
f|{f^{ - 1}}({B_n}):{f^{ - 1}}({B_n}) \to {B_n}
is hereditary shape equivalence,
f
f
may not be a hereditary shape equivalence. Conditions are placed on the
B
n
{B_n}
’s to assure that
f
f
is a hereditary shape equivalence. For example, if
N
f
⊂
⋃
n
=
1
∞
B
n
{N_f} \subset \bigcup \nolimits _{n = 1}^\infty {{B_n}}
, where
B
n
{B_n}
is closed for each
n
=
1
,
2
,
…
,
f
|
f
−
1
(
B
n
)
:
f
−
1
(
B
n
)
→
B
n
n = 1,2, \ldots ,f|{f^{ - 1}}({B_n}):{f^{ - 1}}({B_n}) \to {B_n}
is a hereditary shape equivalence, and
B
n
{B_n}
has arbitrary small neighborhoods whose boundaries miss
⋃
i
=
1
∞
B
i
\bigcup \nolimits _{i = 1}^\infty {{B_i}}
then
f
f
is a hereditary shape equivalence. An immediate consequence is that if
{
B
n
}
n
=
1
∞
\{ {B_n}\} _{n = 1}^\infty
is a pairwise disjoint null-sequence and each
f
|
f
−
1
(
B
n
)
f|{f^{ - 1}}({B_n})
is a hereditary shape equivalence, then
f
f
is a hereditary shape equivalence. Previously G. Kozlowski showed that if
{
f
−
1
(
B
n
)
}
n
=
1
∞
\{ {f^{ - 1}}({B_n})\} _{n = 1}^\infty
is a pairwise disjoint null-sequence and each
f
|
f
−
1
(
B
n
)
f|{f^{ - 1}}({B_n})
is a hereditary shape equivalence, then
f
f
is a hereditary shape equivalence, which can be obtained as an immediate corollary of one of our results.