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The characteristic that defines a perforated state is indeed that it completely surrounds another country. This situation creates a unique geopolitical dynamic, as the surrounding state has the capacity to influence or control the internal affairs of the state enclosed within it. An example of a perforated state is Italy, which completely surrounds the country of San Marino.
In the context of geography, the surrounding state can also face various challenges, such as managing relations with the encapsulated state, dealing with trade routes, and addressing the needs of its own border areas with respect to the enclosed nation. This geopolitical relationship can lead to dependence or tension between the two states.
The other options highlight different concepts: a country separated into multiple regions refers to a fragmented state; a state with no borders with other countries suggests an enclave or landlocked situation; and a region surrounded by territorial waters does not fit the definition of a perforated state, as it relates to maritime boundaries rather than land-based territorial enclosures.