Least Upper Bound: Definition & Subbase

In summary, the conversation discusses the concept of the least upper bound of a family of topologies on a set X. It is defined as the intersection of all topologies that are stronger than each individual topology in the family. The topology generated by the class \bigcup T_i is shown to be the least upper bound, with \bigcup T_i serving as an open subbase. The speaker expresses confusion about why this is the case and is advised to focus on understanding the proof.
  • #1
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I read the following:
"If {[tex]T_i[/tex]} 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 [tex]\bigcup T_i[/tex]; that is, the class [tex]\bigcup T_i[/tex] is an open subbase for the least upper bound of the family {[tex]T_i[/tex]} ."

I understand that the least upper bound L of a family of topologies is the intersection of all topologies which are stronger than each [tex]T_i[/tex] but I don't understand why [tex]\bigcup T_i[/tex] is a subbase for L.
 
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  • #2
Then you should work on the proof of that.
 

1. What is the definition of least upper bound?

The least upper bound, also known as the supremum, is the smallest number that is greater than or equal to all numbers in a given set. In other words, it is the smallest upper bound for a set of numbers.

2. How is least upper bound different from maximum?

The least upper bound is the smallest number that is greater than or equal to all numbers in a set, whereas the maximum is the largest number in a set. The least upper bound may or may not be a member of the set, while the maximum must be a member of the set.

3. What is a subbase for a topological space?

A subbase for a topological space is a collection of sets that can be used to generate the open sets of the space. In other words, any open set in the space can be expressed as a union of sets from the subbase.

4. How is a subbase related to the least upper bound?

A subbase is used to define a topology on a space, and the least upper bound is a concept that is often used in topology. In particular, the least upper bound property is commonly used to prove the existence of open sets in a topological space.

5. Why is the least upper bound important in mathematics?

The least upper bound is an important concept in mathematics because it allows us to define and understand limits, continuity, convergence, and other fundamental concepts in analysis. It also has applications in fields such as economics, physics, and computer science.

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