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.



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    $\begingroup$ 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 $S$. $\endgroup$ – Tunococ Sep 11 '12 at 4:52
  • $\begingroup$ Echoing @Tunococ, if $S$ is a subbasis for the topology $A$, then, by definition, $A = \sigma(S)$. $\endgroup$ – copper.hat Sep 11 '12 at 5:04
  • $\begingroup$ 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... $\endgroup$ – Vien Nguyen Sep 11 '12 at 5:13
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    $\begingroup$ @Vien This seems to be related. $\endgroup$ – Rudy the Reindeer Sep 11 '12 at 5:42
  • $\begingroup$ 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. $\endgroup$ – Rudy the Reindeer Sep 11 '12 at 5:45

Other definitions that are equivalent:

Let $X$ be a topological space with topology $\tau$. A subbase of $\tau$ is usually defined as a subcollection $\mathcal{B}$ of $\tau$ satisfying one of the two following equivalent conditions:

  1. The subcollection $\mathcal{B}$ generates the topology $\tau$.
    [This means that T is the smallest topology containing B: any topology U on X containing B must also contain T.]
  2. The collection of open sets consisting of all finite intersections of elements of $\mathcal{B}$, together with the set $X$ and the empty set, forms a basis for $\tau$.
    [This means that every non-empty proper open set in T can be written as a union of finite intersections of elements of $\mathcal{B}$. Explicitly, given a point x in a proper open set $U$, there are finitely many sets $S_1, \dots, S_n \in \mathcal{B}$, 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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