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I have a question about the following definition:
A subbasis $S$ for a topology on a set $X$ is a collection of subsets of $X$ whose union equals $X$. The topology generated by the subbasis $S$ is defined to be the collection $T$ of all unions of finite intersections of elements of $S$.
If you have a subbasis $S$ for a topology $A$, then is the topology generated by $S$ necessarily also $A$? It seems like you could have many different subbases for $A$, but my intuition is that they might not all generate the same topology $A$. Is there something i'm missing? Thanks for any help/clarification.

Sincerely,

Vien

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The definition you gave starts from the subbasis $S$, then generates the topology $T$ from it. If you say that $S$ is a subbasis of $A$, it means $A$ is a topology, and it is generated by $S$. –  Tunococ Sep 11 '12 at 4:52
    
Echoing @Tunococ, if $S$ is a subbasis for the topology $A$, then, by definition, $A = \sigma(S)$. –  copper.hat Sep 11 '12 at 5:04
    
thank you guys for your answers. Funny thing, as i look through the internet, i'm finding a couple slightly different looking definitions of subbasis from the one in my book... –  Vien Nguyen Sep 11 '12 at 5:13
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@Vien This seems to be related. –  Rudy the Reindeer Sep 11 '12 at 5:42
    
Maybe you could list the different definitions in your question (or possibly post them as an answer). It would be beneficial for both future readers of this thread and yourself. –  Rudy the Reindeer Sep 11 '12 at 5:45

1 Answer 1

up vote 1 down vote accepted

Other definitions that are equivalent:

Let X be a topological space with topology T. A subbase of T is usually defined as a subcollection B of T satisfying one of the two following equivalent conditions: The subcollection B generates the topology T. This means that T is the smallest topology containing B: any topology U on X containing B must also contain T. The collection of open sets consisting of all finite intersections of elements of B, together with the set X and the empty set, forms a basis for T. This means that every non-empty proper open set in T can be written as a union of finite intersections of elements of B. Explicitly, given a point x in a proper open set U, there are finitely many sets S1, …, Sn of B, such that the intersection of these sets contains x and is contained in U. (wiki)

A collection of subsets of a topological space that is contained in a basis of the topology and can be completed to a basis when adding all finite intersections of the subsets. (wolfram mathworld)

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