Base versus basis

This is a question on the borderline between mathematics and English. I wonder in mathematics, are there some general differences between concepts with base and with basis in their names? In other words, do you have some idea when to use which, because I often use one when the other is expected.

For example, a basis of a vector space and a base of a topology.

Thanks and regards!

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Admittedly both words are quite overloaded... – J. M. Oct 31 '11 at 17:49
J.M. probably means having many different meanings – user12205 Oct 31 '11 at 17:59
"All your basis" – The Chaz 2.0 Oct 31 '11 at 18:17
@Tim: It's a humourous reference to this. – joriki Oct 31 '11 at 18:24
I was in the middle of making a joke when my iPhone prematurely submitted the comment. Then I lost Internet connection! But joriki figured it out :) – The Chaz 2.0 Oct 31 '11 at 18:30