Let K be a locally compact
metric space. An isotopy on K is a continuous family of homeomorphisms
ht: K → K for t ∈ I such that h0= id. Let (K) denote the space of isotopies of K
with C − 0 topology. Conjecture: Let X be metric and Y a closed subset of X. Then
every map f : Y →ℐ(K) can be continuously extended to X. The conjecture is
proved for the following cases: (1) K is a l-complex, (2) K is compact and X is
finite-dimensional, (3) K is compact, Y is compact and finite-dimensional, and X is
separable, and (4) Y is of type 1 in a compact space X. Y is of type 1 in X if the
closure of the set of points of X which do not have a unique closest point in Y does
not intersect Y .
