According to PEP-249, on connection closing all the related cursors must become no-op as well. The current implementation of closes and checks has an issue at this point.

Connection.close() only sets the connection attribute to True:

While the Cursor checking method is only looking for the Cursor._closed attribute:

That means cursor don't know if the related connection is already closed. Technically, it doesn't look like a big problem, as Cursor.execute() delegates to Connection, and the connection will raise an exception already closed. But there are several commands, which will be runned on a closed Cursor before the exception raised, for example, classify_stmt(sql).

To avoid that it'll be good to add into Cursor._raise_if_already_closed() a line like:

if self._connection.is_closed:
    raise ...

Checking a bool is much more faster than regexping SQL code and passing it into a connection method.

Plus to this, I'd make is_closed attribute public, so that users could check if a cursor/connection is closed.