Granas and Dugundji obtained the following generalization of the Knaster-Kuratowski-Mazurkiewicz theorem. Let
X
X
be a subset of a topological vector space
E
E
and let
G
G
be a set-valued map from
X
X
into
E
E
such that for each finite subset
{
x
1
,
…
,
x
n
}
\{ {x_1}, \ldots ,{x_n}\}
of
X
,
c
o
{
x
1
,
…
,
x
n
}
⊂
∪
i
=
1
n
G
x
i
X,co\{ {x_1}, \ldots ,{x_n}\} \subset \cup _{i = 1}^nG{x_i}
and for each
x
∈
X
,
G
x
x \in X,Gx
is finitely closed, i.e., for any finite-dimensional subspace
L
L
of
E
,
G
x
∩
L
E,Gx \cap L
is closed in the Euclidean topology of
L
L
. Then
{
G
x
:
x
∈
X
}
\{ Gx:x \in X\}
has the finite intersection property. By relaxing, among others, the condition that
X
X
is a subset of
E
E
, we obtain a further generalization of the theorem and show some of its applications.