Bases and subbases questions in point set topology I have some questions regarding point set topologies.  I know if one is given a topology, you can extract the base for a topology, also, if given two identical bases, they can generate the same topology.
but are the following possible
If I am given two identical bases $B_1=B_2$, can $B_1$ generate a topology different from $B_2$.
Likewise, if given two non identical bases, is it true sometimes that the two different bases 
My other question are the same as the above but for the case of subbase.
Thank you in advance.
Seth
 A: There is not a standard way to extract a base (or subbase) for a given topology. In fact often the direction is the other way around: a topology is first introduced by giving a base for it (like in the case of metric spaces, where the open balls form a standard base, or for ordered spaces, where the sets $U(a) = \{ x \in X \mid x > a \}$ and $L(a) = \{ x \in X \mid x < a \}$ for $a \in X$ form a subbase).
By definition, the topology generated by a subbase or base is the smallest topology on the set that contains that subbase or base. This is a uniquely defined topology (it's the intersection of all topologies that contain it, and the discrete topology is always one of those) so equal (sub)bases give equal topologies.
As mentioned, a given topology in general will have many different bases or subbases that generate it. The open balls (metric base) vs. open rectangles (product topology base) for the plane are classical examples of that, but there are more trivial ones as well (the topology itself is a base for itself, and if $X$ is $T_1$, so is the topology minus $X$ itself, e.g.)
A: For identical bases to generate different topologies, there would have to be some further ingredient in the definition of "to generate" that might account for the difference. For instance, identical elements of a set might generate different subgroups if different group operations are defined on the elements. However, in the definition of what it means for a base to generate a topology, there are no further ingredients; the topology is entirely determined by the base, namely as the set of all unions of elements of the base. Thus identical bases generate identical topologies.
A: To expand on nullUser's comment (for the case of subbases):
If $S$ is a subbase of a topology $T$ it means that, by definition of subbase, that $T$ is the smallest topology such that $S \subset T$.
Hence, if $S = S'$ then $T = T'$ since the "smallest thingamajig containing $S$" is the intersection of all thingamajigs containing S.
For the other direction: If both $S$ and $S'$ are subbases of $T$ it follows immediately that they both generate $T$.
