# The difference between and basis and subbasis for a topology

• December 6th 2011, 10:58 PM
Jame
The difference between and basis and subbasis for a topology
The two definitions seem very similar to me and is confusing.

To express it simply

a basis generates a topology T and a subbasis generates a basis for a topology?

Is there an example that could clarify this difference?

• December 6th 2011, 11:06 PM
Drexel28
Re: The difference between and basis and subbasis for a topology
Quote:

Originally Posted by Jame
The two definitions seem very similar to me and is confusing.

To express it simply

a basis generates a topology T and a subbasis generates a basis for a topology?

Is there an example that could clarify this difference?

Roughly, bases are collections of subsets by which all open sets can be obatined via unions from that family and subbases are collection of open subsets for which any open subset can be obtained by unions of INTERSECTIONS of elements of the collection. For example the set of all infinite rectangles $\{\mathbb{R}\times (a,b):a,b\in\mathbb{R}\}$ forms a subbasis for the usual topology on $\mathbb{R}^2$ but not a basis. Explain to me why.
Oh i see. There are open sets in $\mathBB{R}^2$ which we cannot get from unioning these infinite rectangles together, but they can be made by unioning intersects of these rectangles.