Easy question regarding the basis for a topology

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    Basis Topology
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SUMMARY

A basis for a topology \( T \) on a set \( X \) is defined as a collection \( B \) of open sets such that every open set in \( T \) can be expressed as a union of sets from \( B \). The discussion confirms that taking the set of all unions of sets in \( B \) indeed reconstructs the topology \( T \). This conclusion is derived directly from the definition of a topology, which is characterized by its open sets being closed under unions and finite intersections.

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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 $T$ ?
 
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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.
 

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