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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.2012-10-01

3 Answers 3