Easy question regarding the basis for a topology

[christian1357]

Hello, I know that given a set $X$ and a topology $T$ on $X$ that a basis $B$ for $T$ is a collection of open sets of $T$ such that every open set of $T$ is the Union of sets in $B$. My question is: does taking the set of all Unions of sets in $B$ give exactly the topology $T$ ?

 

[mathman]

Yes - directly from the definition. A topology is defined by open sets. The set of all open sets is closed under all unions and finite intersections, so adding more open sets to the basis doesn't change the topology.