Let
X
1
←
X
2
←
⋯
{X_1} \leftarrow {X_2} \leftarrow \cdots
be an inverse sequence of spaces and maps satisfying (i) each
X
n
{X_n}
has the homotopy type of a CW complex, (ii) each
f
n
{f_n}
is a Hurewicz fibration, and (iii) the connectivity of the fiber of
f
n
{f_n}
goes to
∞
\infty
with
n
n
. Let
X
^
\hat X
be the inverse limit of the sequence. It is shown that the natural homomorphism
H
ˇ
k
(
X
^
,
G
)
→
H
k
(
X
^
,
G
)
\check {H}^k(\hat {X},G) \to H^k(\hat {X}, G)
(from Čech cohomology to singular cohomology, with ordinary coefficient module
G
G
) is an isomorphism for all
k
k
. It follows that
lim
→
n
[
X
n
,
K
(
G
,
k
)
]
≅
[
X
^
,
K
(
G
,
k
)
]
{\lim _{ \to n}}[{X_n},K(G,k)] \cong [\hat X,K(G,k)]
for any Eilenberg- Mac Lane space
K
(
G
,
k
)
K(G,k)
. It is also shown that, except in trivial cases,
X
X
does not have the homotopy type of a CW complex.