Uniqueness of Borel sigma algebra on X

Question

I have a question about the uniqueness of $\mathcal{B}(X)$, Borel sigma algebra on X. Suppose $A\subseteq X$, then is $\{\varnothing, A, A^c, X\}$ a Borel sigma algebra on X?

Answer 1

The Borel $\sigma$-algebra is defined in terms of a topology on $X$. Indeed, it's the smallest $\sigma$-algebra containing all the opens. That is, if $\tau$ is a topology on $X$, then $\mathcal{B}(X,\tau)$ is the (unique) smallest $\sigma$-algebra containing $\tau$.

So to ask what the Borel algebra on $X$ is, without first fixing a topology on $X$, is a meaningless question.

That said, there are topologies on $X$ for which $\{\emptyset, A, A^c, X\}$ is the whole Borel algebra. Can you find one? As a hint, you shouldn't have to change much...

As one last aside, you might ask if every $\sigma$-algebra on a set $X$ is the Borel algebra for some topology on $X$. It turns out (surprisingly, I think!) that the answer is "no". See here for more information.

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I hope this helps ^_^