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I'm confused about subbases: the sub in the name suggests that a subbase $S$ is a subset of a base $B$ of a topology $T$.

Can there be a topology $T$ such that it is generated by a subbase that is not a subset of a given base $B$ that generates $T$?

The definitions are:

A subbase $S$ is a subset of a topology $T$ that generates $T$, i.e. $T$ is the smallest topology such that $S \subset T$.

A base $B$ is a subset of a topology $T$ that generates $T$ and such that every set in $T$ can be written as a union of elements in $B$.

Is this correct?

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A subbase can always be enlarged to a basis for a topology, but that's not saying much since the topology $T$ is a basis for itself. –  Grumpy Parsnip Apr 28 '12 at 16:41
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up vote 4 down vote accepted

Given a topology, there's typically lots of bases for the topology. For example, in $\mathbb{R}^2$ with the usual topology, open rectangles parallel to the axes are a base for the topology, but so are open disks or open triangles. These bases are all "compatible" in the sense that they generate the same topology, but of course they are different as subsets of the collection of open subsets of $\mathbb{R}^2$.

When you say that $S$ is a subbase, you typically have a particular base $B$ in mind. In the example above, open rectangles with width at least 10 are a subbase with respect to the base with all open rectangles (cause you can get all open rectangles by intersecting these). But they're not a subbase with respect to the base with all open circles, because they're not even a subset of that collection.

(EDIT - to be clear: I am making an ad hoc definition in the previous paragraph, saying that $S$ is a subbase "with respect to" $B$ if $S$ is contained in $B$.)

In practice, this distinction rarely matters. The point is that if I give you a random set $X$ and you want to put a topology on it, you can pick a collection $S$ of the subsets of $X$ and call those sets open. That won't be a topology, but it will generate one. In general $S$ will be a subbase with respect to some base $B$ of that topology, maybe you can explicitly describe $B$, or maybe you can't, depends what $S$ you chose.

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Thank you very much for this clear answer! This answers my question very very well! –  Matt N. Apr 28 '12 at 17:30
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