Least Upper Bound: Definition & Subbase

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SUMMARY

The least upper bound (LUB) of a family of topologies {T_i} on a set X is defined as the topology generated by the union of the topologies, denoted as \bigcup T_i. This union serves as an open subbase for the least upper bound, which is the intersection of all topologies stronger than each T_i. Understanding this relationship is crucial for grasping the foundational concepts of topology and its applications in mathematical analysis.

PREREQUISITES
  • Basic knowledge of topology concepts, including open sets and topologies.
  • Familiarity with the definitions of least upper bounds and subbases.
  • Understanding of set theory, particularly unions and intersections.
  • Experience with mathematical proofs and logical reasoning.
NEXT STEPS
  • Study the proof that \bigcup T_i is a subbase for the least upper bound L.
  • Explore the implications of the least upper bound in various topological spaces.
  • Investigate examples of families of topologies and their least upper bounds.
  • Learn about stronger topologies and their relationships to subbases.
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Mathematicians, students of topology, and anyone interested in advanced mathematical concepts related to topological spaces and their properties.

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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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