The above paragraph has some nice implications. You probably heard of "Group Theory" before. A group

is a set with an operation

so that: (i)

for any

(ii)

for all

(iii) there is an "identity element"

so that

(iv) for any element

there is an "inverse"

so that

. For example, the integers

are a group under the operation

. Now if we let

be the set of all Mobius transformation on the Riemann Sphere then

is a group under

(function composition). Note, we could have not made such a statement about a group if the Mobius transformations were defined only on

because as stated above we cannot necessarily compose them. If you want you can try proving that

, the "Mobius group", is actually a group.