  "A collection of interconnected networks" defines what the Internet is, but
not what it is not. Events threaten Internet fragmentation: politics suggest
countries or ISPs may secede or be de-peered, disputes between ISPs result in
persistent unreachability between their customers, and architectural changes
risk breaking the "one" Internet. Understanding such threats benefits from a
testable definition of what the Internet is and is not, enabling discussion and
quantification of partial connectivity. We provide a conceptual definition
giving an idealized asymptote of connectivity. It implies peninsulas of
persistent, partial connectivity, and islands when one or more computers are
partitioned from the main Internet. We provide algorithms to measure,
operationally, the number, size, and duration of peninsulas and islands. We
apply these algorithms in rigorous measurement from two complementary
measurement systems, one observing 5M networks from a few locations, and the
other a few destinations from 10k locations. Results show that peninsulas
(partial connectivity) are about as common as Internet outages, quantifying
this long-observed problem. Root causes show that most peninsula events (45%)
are routing transients, but most peninsula-time (90%) is from a few long-lived
events (7%). Our analysis helps interpret DNSmon, a system monitoring the DNS
root, separating measurement error and persistent problems from underlying
differences and operationally important transients. Finally, our definition
confirms the international nature of the Internet: no single country can
unilaterally claim to be "the Internet", but countries can choose to leave.
