# question about bases of a given topology

Given a set X with a given topology T, can T have more than one basis that generates T? Can you explain your answer?

(I don't think that it can, but I can't think of why not either).

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Yes, there can be several different bases for a given topology. For example, on $\Bbb R$, we can generate the usual topology via open intervals of rational length, open intervals of irrational length, etcetera.

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It is quite possible have many different bases for a topology. For example, consider the real line $\mathbb{R}$ with the usual topology. This topology is generated by any of the following bases:

• The collection of all open intervals $(a,b)$ with $a<b$,
• The collection of all open intervals $(p,q)$ where $p$ and $q$ are rationals,
• The collection of all open intervals $(p,q)$ where $p$ and $q$ are irrrationals.

There are many other possibilities

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