Least Upper Bound: Definition & Subbase

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The least upper bound (L) of a family of topologies {T_i} on a set X is defined as the intersection of all topologies stronger than each T_i. The discussion centers on the assertion that the union of the topologies, ∪T_i, serves as an open subbase for L. Understanding this requires proving that any open set in L can be generated from the open sets in ∪T_i. The proof involves demonstrating that the topology generated by ∪T_i includes all necessary open sets to form L. Clarifying this relationship is essential for grasping the concept of least upper bounds in topology.
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I read the following:
"If {T_i} is a non empty family of topologies on our set X, then the least upper bound of this family is precisely the topology generated by the class \bigcup T_i; that is, the class \bigcup T_i is an open subbase for the least upper bound of the family {T_i} ."

I understand that the least upper bound L of a family of topologies is the intersection of all topologies which are stronger than each T_i but I don't understand why \bigcup T_i is a subbase for L.
 
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Then you should work on the proof of that.
 

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