Topology: Finite Complement & Defining Limit Points

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In summary, the conversation discusses the conditions for defining a topology, specifically the finite complement topology, and determining when a point is a limit of a subset in X. The only case to consider is when X is infinite, and there are two subcases: A is finite and A is infinite. In the former case, A has no limit points, while in the latter case, every point is a limit point. This can be proven by considering the complement of A in X and using the fact that X/U is finite for any open subset U.
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doodlepin
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Question is: X is a set and tau is a collection of subsets O of X such that X - O is either finite or all of X. Show this is a topology and completely described when a point x in X is a limit of a subset A in X.

I already proved that this satisfies the conditions for defining a topology (called the finite complement topology) But I am having a lot of trouble with defining when a point is a limit. I know that any open subset (defined by tau) containing such a limit point must have a non-empty intersection with A for it to be a limit point.

I have tried considering multiple cases: I know if X is finite then it is the discrete topology and therefore no limit points exist. So the non trivial case if when X is infinite.
Now, if A is finite then for any open subset containing x, O, the complement with A would obviously be finite so therefore non empty?
I'm sort of stuck and could use a nudge in some helpful direction.
 
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  • #2
You are correct in saying that the only case you need to find out is when X is infinite.
Now there are two more cases to consider:
1) A is finite: I claim that A has no limit points in this case. Hint: consider X/A
2) A is infinite: I claim that every point is a limit point in this case. Take U open, then X/U is finite. So it can not happen that [tex]A\subseteq X\setminus U[/tex]. I'll let you complete the proof...
 
  • #3
wow you are totally right. Thank you so much. Sometimes you start thinking too hard about something and you start to over analyze it. :)
 

1. What is a finite complement topology?

A finite complement topology is a type of topology in which the open sets are all possible complements of a finite set of points in a given space. This means that the closed sets are all finite sets or the entire space itself.

2. How is a finite complement topology different from other topologies?

A finite complement topology is different from other topologies in that it is not based on open or closed intervals, but rather on finite sets. This makes it a discrete topology, meaning that every subset is both open and closed.

3. What is a limit point in topology?

In topology, a limit point is a point in a given set that can be approximated by other points in the set. In other words, any open set containing the limit point must also contain other points in the original set. This is important in understanding the structure and properties of topological spaces.

4. How are limit points defined in a finite complement topology?

In a finite complement topology, a limit point is defined as any point in the given space that is not in the finite set of points defining the topology. This means that every open set containing the limit point must also contain infinitely many points from the original set.

5. What are some real-world applications of finite complement topology?

Finite complement topology has applications in various fields such as computer science, physics, and economics. In computer science, it is used to study algorithms and data structures. In physics, it is used to model phase transitions and energy landscapes. In economics, it is used to analyze market trends and trading strategies.

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