Define a new topology on the reals

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In summary, the author is verifying that taking \mathbb{R}, the empty set, and finite sets as closed results in a topology. The author argues that the empty set is finite and therefore closed, and that the union and intersection of finite sets is again finite and therefore closed. The author also mentions that \mathbb{R} is defined as closed in the statement of the question.
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andlook
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Homework Statement


Verify that taking [tex]\mathbb{R}[/tex], the empty set and finite sets to be closed gives a topology.

Homework Equations





The Attempt at a Solution



Clearly the empty set is finite as it has 0 elemnts, and so is closed.

If [tex] X_i [/tex], for i= {1,...,n}, are finite sets then clearly the union of finitely many finite sets is again finite and so is closed.

Let [tex]|X_i| = m_i [/tex], where [tex]m_i[/tex] is the number of elements in [tex]X_i[/tex], then [tex]|\bigcap X_i |[/tex] is at most max{[tex]m_i[/tex]} or at least 0 if the intersection is empty. Either way it is again finite and so is defined as closed.


I think I have shown the empty set, arb unions, arb intersections are closed in this topology, but I can't see how [tex] \mathbb{R}[/tex] could be included...
Thanks
 
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  • #2
andlook said:
I think I have shown the empty set, arb unions, arb intersections are closed in this topology, but I can't see how [tex] \mathbb{R}[/tex] could be included...
Reread the description of the closed sets. It says:
1) That all finite sets are closed.
2) That the empty set is closed (this is really redundant, but there's no harm in including it).
3) That [itex]\mathbb{R}[/itex] is closed.
 
  • #3
Of course, as defined in the statement of the question!

Thanks
 

1. What is topology?

Topology is a branch of mathematics that studies the properties of space and the relationships between different objects within that space. It focuses on the concept of "closeness," rather than traditional measures of distance.

2. How is topology related to the reals?

The reals, or the set of real numbers, can be thought of as a one-dimensional space. Topology allows us to define a structure on this space by specifying which sets of real numbers are considered "open" or "closed." This structure can then be used to study the properties and relationships of real numbers.

3. Why would one want to define a new topology on the reals?

A new topology on the reals can provide a different perspective and understanding of the relationships between real numbers. It can also be useful in solving certain mathematical problems or in studying specific types of functions or spaces.

4. How is a new topology defined on the reals?

A new topology on the reals is defined by specifying a set of open sets that satisfy certain properties, such as being closed under finite intersections and unions. These open sets determine which sets of real numbers are considered "close" to each other in the new topology.

5. Can a new topology on the reals change the fundamental properties of real numbers?

No, a new topology on the reals does not change the fundamental properties of real numbers, such as their ordering or arithmetic operations. It simply provides a new way of looking at these properties and studying them in a different context.

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