Limit Points of Sets: Find Interior, Boundary & Open/Closed

In summary, the question is to find the limit points, interior points and boundary points of the set {(x,y)|(x,y)=(1/n,1-1/n), where n is a positive integar}. There is confusion about whether 0 and 1 are boundary points or limit points. However, as the set is in E^2, all points in it must be of the form (a, b), an ordered pair of numbers, and cannot be a single number. Therefore, both you and your mate need to reassess the problem.
  • #1
reddevils78
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Homework Statement



Consider the set in E^2 of points {(x,y)|(x,y)=(1/n,1-1/n), where n is a positive integar}. Find the limit points, interior points and boundary points. Determine whether this set is open or closed.

Homework Equations





The Attempt at a Solution


I figured, 0,1 must be the boundary points of the set but a mate claims they are the limit points instead that has brought me into this confusion of what exactly is the difference between the two.
 
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  • #2
"0" and "1" can't be boundary points of the set. This set is in R2- all points in it, all boundary points, all limit points, etc. must be of the form (a, b), an ordered pair of numbers, not a number. You and your mate both need to rethink the entire problem!
 

What are limit points of a set?

Limit points of a set are points that are either contained within the set or are arbitrarily close to the set. They are also referred to as accumulation points or cluster points.

How do you find the interior of a set?

The interior of a set is the largest open subset of the set. It can be found by taking all the points in the set that are not limit points and then removing them from the set.

What is the boundary of a set?

The boundary of a set is the set of all limit points of the set. It is the boundary between the interior and exterior of the set.

How can you determine if a set is open or closed?

A set is open if all of its points are interior points, meaning that no points on the boundary are included in the set. A set is closed if it contains all of its limit points.

What is the relationship between limit points, interior, boundary, open, and closed sets?

Limit points are points that are either contained within a set or are arbitrarily close to the set. The interior of a set is the largest open subset of the set and does not contain any limit points. The boundary of a set is the set of all limit points. A set is open if all of its points are interior points, and closed if it contains all of its limit points.

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