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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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up vote 2 down vote accepted

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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Every topology is its own basis. Unlike linear algebra there is no requirement for minimality in any sense. The entire topology is always a basis, and in many of the topologies we meet in our every day lives there are natural bases to begin with, so there are usually many bases to a topology.

For example the trivial topology has only itself as a basis, so it's unique. On the other hand, the discrete topology on a space with more than two points would have several bases, take all the singletons and add or remove addiional arbitrary sets.

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