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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

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Certainly the standard topology in $\mathbb{R^n}$ can be generated by open balls or open rectangles (these are different when $n\geq 2$, so two different bases can give the same topology. The other statement is false, "can $B_1$ generate a topology different from $B_2$." since $B_1=B_2$ this is "can $B_1$ generate a topology different from $B_1$" which is clearly false. –  nullUser Oct 1 '12 at 20:55

3 Answers 3

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.)

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I think my source of confusion derives from this question: math.stackexchange.com/questions/63143/… –  Seth Mai Oct 1 '12 at 22:05
    
I first learn bases ann neighborhood bases through a different textbook, and then when i start taking point set topology using Munkres' text, my prof just use base and sub bases but did not go further. I know all these definitions are equivalent, but the impression from the question (the link mentioned) are that one can have two different topologies from the same base. So in essence, when one talks about neighborhood base, can one talk about same results derived from neighborhood bases and transfer verbatim to just bases. –  Seth Mai Oct 1 '12 at 22:12

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 $T$.

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Sorry when you said "smallest thingamajig containing S" did you meant "smallest thing containing S". I am trying to guess through your typos. Thanks in advance –  Seth Mai Oct 1 '12 at 21:59
    
@SethMai Well, yes, you can just as well replace "thingamajig" with "gizmo" or "thing". Which "typos" are you referring to? If you point them out I'll correct them. –  Matt N. Oct 2 '12 at 8:29
    
I did not understand what you meant when you wrote "thingamajig" I thought it was a typo on your part. –  Seth Mai Oct 2 '12 at 12:22

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.

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