Abstract
AbstractLet $$K=2^\mathbb {N}$$
K
=
2
N
be the Cantor set, let $$\mathcal {M}$$
M
be the set of all metrics d on K that give its usual (product) topology, and equip $$\mathcal {M}$$
M
with the topology of uniform convergence, where the metrics are regarded as functions on $$K^2$$
K
2
. We prove that the set of metrics $$d\in \mathcal {M}$$
d
∈
M
for which the Lipschitz-free space $$\mathcal {F}(K,d)$$
F
(
K
,
d
)
has the metric approximation property is a residual $$F_{\sigma \delta }$$
F
σ
δ
set in $$\mathcal {M}$$
M
, and that the set of metrics $$d\in \mathcal {M}$$
d
∈
M
for which $$\mathcal {F}(K,d)$$
F
(
K
,
d
)
fails the approximation property is a dense meager set in $$\mathcal {M}$$
M
. This answers a question posed by G. Godefroy.
Funder
University College Dublin
Publisher
Springer Science and Business Media LLC
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