TITLE: question about bases of a given topology
QUESTION [5 upvotes]: Given a set $X$ with a given topology $T$, can $T$ have more than one basis that generates $T$? Can you explain your answer?
(I don't think that it can, but I can't think of why not either).

REPLY [2 votes]: Every topology is its own basis. Unlike linear algebra there is no requirement for minimality in any sense. The entire topology is always a basis, and in many of the topologies we meet in our every day lives there are natural bases to begin with, so there are usually many bases to a topology. 
For example the trivial topology has only itself as a basis, so it's unique. On the other hand, the discrete topology on a space with more than two points would have several bases, take all the singletons and add or remove addiional arbitrary sets.