An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space ( the closure of "'N "'in ( " ", " ) is "'N "'* " { " } .)
12.
Note that any member of the modular group maps the projectively extended real line one-to-one to itself, and furthermore bijectively maps the projectively extended rational line ( the rationals with infinity ) to itself, the irrationals to the irrationals, the transcendental numbers to the transcendental numbers, the non-real numbers to the non-real numbers, the upper half-plane to the upper half-plane, et cetera.
13.
In each case the absolute value | " f " / " g " | approaches + ", and so the quotient " f " / " g " must diverge, in the sense of the extended real numbers . ( In the framework of the projectively extended real line, the limit is the unsigned infinity " in all three cases . ) Similarly, any expression of the form " a " / 0, with ( including and ), is not an indeterminate form since a quotient giving rise to such an expression will always diverge.
