The usual topology is the smallest topology containing the upper and lower topology

  • Thread starter rlkeenan
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In summary, the usual topology for R, denoted by TR(R), is the smallest topology containing the lower topology (Tl) and the upper topology (Tu). This topology is generated by sets of the form (-∞, b) and (a, ∞) and can be shown to be equal to both Tl and Tu.
  • #1
rlkeenan
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Trying to prove:
The usual topology is the smallest topology for R containing Tl and Tu.
NOTE: for e>0
The usual topology: TR(R)={A<R|a in A =>(a-e,a+e)<A}
The lower topology: Tl(R)={A<R|a in A =>(-∞ ,a+e)<A}
The upper topology: Tu(R)={A<R|a in A =>(a-e, ∞)<A}

3. The Attempt at a Solution
claim 1: If T is a topology for R s.t. Tl<T and Tu<T then TR<T
proof: let Tl<T and Tu<T
claim 1.1: If p is in TR then p is in T
proof: let p=(a,b) for any a,b in R
Then p is in TR by definition of TR
We know (-∞,b) is in Tl<T and (a,∞) is in Tu<T therefore (-∞,b),(a,∞) are in T
and (-∞,b)^(a,∞)=(a,b) is in T since T is a topology
therefore p is in T
therefore TR<T
...not sure where to go after here maybe show that if there is a T'<TR s.t. Tu<T' and Tl<T' then T'=TR
 
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  • #2


The smallest topology containing the two would be generated by the intersections of sets open in the two topologies , i.e., sets of the form (-∞ ,b)
and (a, ∞).This is either empty or just the interval (a,b).
 
  • #3


I get that but that implies that TR={(a,b)|a,b in R} so you get TR<Tl and TR<Tu and we already know that Tl<TR and Tu<TR so then you have that TR=Tl and TR=Tu so then you have Tl=Tu which can't be right
 

1. What is the usual topology?

The usual topology is a type of topology in mathematics that is defined as the smallest topology containing the upper and lower topology. It is commonly used in functional analysis and topology to study continuous functions and convergence.

2. How is the usual topology defined?

The usual topology is defined as the smallest topology that contains the upper and lower topology. This means that it includes all the open sets from the upper and lower topology, as well as any other open sets that can be obtained by taking unions and finite intersections of these sets.

3. What are the upper and lower topologies?

The upper and lower topologies are two specific types of topologies that are used to define the usual topology. The upper topology consists of all open sets that contain a point x, while the lower topology consists of all open sets that are contained in a point x. These two topologies are used to define the usual topology because they represent the most basic open sets that contain or are contained in a point x.

4. Why is the usual topology important?

The usual topology is important because it allows us to study the properties of continuous functions and convergence in a general and abstract way. It also helps us to define and understand other types of topologies that are commonly used in mathematics, such as the product topology and the subspace topology.

5. How is the usual topology used in mathematics?

The usual topology is used in various areas of mathematics, including functional analysis, topology, and real analysis. It is used to study continuous functions and convergence, as well as to define and understand other types of topologies. It also has applications in fields such as physics and economics, where it is used to model and analyze continuous systems and processes.

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