When it comes to the network, a closed-loop network refers to the fact that the network operator is also the issuing bank (e.g. Amex, Discover). Open loop networks, the network operator is not the issuing bank (e.g. Visa, Mastercard).

Patio11 is correct in that from a gift card usage perspective, Amex is open loop. But I'm just pointing out that from a network perspective, they are not.