What are the differences between derived and closure points in sets?

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In summary, the conversation is about understanding sets with derived points that are different from closure points, and the concept of a base. One example given is the set {1/n : n in Naturals}, which has a D-pt of {0} and closure pts of {all set}. The terminology used is not standard and the speaker is asked to provide definitions for clarity.
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Unassuming
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Homework Statement


I am looking for examples of sets that have derived pts that are different from closure pts because I am trying to understand them better.

Also, if you can , please try to bring the word "base" into this. I do not understand quite fully a base. I know the definition and I know it is similar to a generator. (I hope)


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The Attempt at a Solution



I know that {1/n : n in Naturals} has D-pt {0} and closure pts {all set}
 
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  • #2
I'm not sure you are using very standard terminology. Can you state the definitions of your words? I would say the closure of S={1/n} is S union {0}.
 
  • #3
Your right, I'm sorry. My closure was wrong
 

FAQ: What are the differences between derived and closure points in sets?

1. What is the difference between derived points and closure points?

Derived points are points that are obtained by taking the limit of a sequence of points in a topological space. Closure points are points that are in the closure of a set, meaning they cannot be separated from the set by open sets. In simpler terms, derived points are obtained from a sequence of points, while closure points are obtained from a set.

2. Can a point be both a derived point and a closure point?

Yes, a point can be both a derived point and a closure point. This is because a point can be in the limit of a sequence of points and also in the closure of a set.

3. How are derived points and closure points related?

Derived points and closure points are related in that they both involve the concept of limit. Derived points are obtained by taking the limit of a sequence of points, while closure points are obtained by taking the limit of a set. In some cases, a derived point can also be a closure point.

4. What is the importance of derived points and closure points in topology?

Derived points and closure points are important in topology because they help define the behavior of points in a topological space. They also give insight into the structure and characteristics of a space, such as its connectedness and compactness.

5. How do derived points and closure points relate to open and closed sets?

Derived points and closure points are closely related to open and closed sets. In fact, a point is a derived point if and only if it is in the closure of every open set containing it. Similarly, a point is a closure point if and only if it is in the intersection of all closed sets containing it.

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