Does the sigma algebra generated by all the open sets same as that generated by all the basic open sets? Let $(X,\tau)$ be a topological space. Does the sigma algebra generated by all the open sets same as that generated by all the basic open sets?
 A: Consider the real line with the discrete topology: each subset is open hence the $\sigma$-algebra generated by them is the power set. 
If we consider the basic open set $(\{x\},x\in \mathbb R)$, then the $\sigma$-algebra generated by them is the collection of sets $S$ such that either $S$ is countable or $\mathbb R\setminus S$ is countable. 
A: Please delete this answer, which duplicates Davide's.
No, they may not be the same. 
Consider an uncountable set $X$ with the discrete topology. Every set is open, so the $\sigma$-algebra generated by the open sets is the set of all subsets of $X$.
A basis for this topology is given by the set of single-element sets. The $\sigma$-algebra generated by these is the set of all sets that are either countable or have countable complement. 
To prove this assertion, note that a countable set is a countable union of single-element sets. Thus it's enough to check that the described collection of subsets is in fact a $\sigma$-algebra. This isn't difficult, since a countable union of countable sets is countable.
A: No in general (see the answer of Davide). 
The answer is yes if you are dealing with a countable basis. Then any open set is a union of a countable collection of basis-elements, so if these belong to the $\sigma$-algebra then so does the open set.
