Exploring Finite Sets & Limit Points

In summary, in any metric space, a finite set does not have any limit points. This is because there exists a "shortest" distance between points, and taking a smaller distance than this shows that the set cannot have any limit points. However, this is not always true for topological spaces, as there are examples where every point is a limit point. But in T1 topological spaces, where every two distinct points have a separating open set, this statement holds true.
  • #1
dalcde
166
0
Is it true that finite sets don't have limit points?
 
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  • #2
it depends on what topology u use on the unversal set
 
  • #3
The real numbers and the Euclidean metric.
 
  • #4
Yes, in any metric space, a finite set has no limit points. A point p is a limit point of set A if and only if, for any [itex]\delta> 0[/itex], there exist a point, q, of A other than p such that [itex]d(p,q)< \delta[/itex]. If A is a finite set, then there exist a "shortest" distance between points: M= min(d(p,q)) where the minimum is take over all pairs of points in A. Taking [itex]\delta[/itex] to be smaller than M shows that A cannot have any limit points.
 
  • #5
with the eucliden metric on R we can deduct the standard topology on R which is a Hausdorff space so the set of limit points is close in R
 
  • #6
Not true in general for topological spaces. For example, for any set X with the indiscrete topology [itex]\{\emptyset,X\}[/itex], every point is a limit point of every set with at least two elements.

But it's true for every T1 topological space. This means that for any two distinct points x and y, there is an open set containing x but not y.
 
  • #7
Thanks!
 

What is a finite set?

A finite set is a set that contains a limited or countable number of elements. This means that the set has a specific number of distinct objects, and that number is not infinite. For example, the set {1, 2, 3, 4, 5} is a finite set because it contains 5 elements.

What is an infinite set?

An infinite set is a set that contains an unlimited or uncountable number of elements. This means that the set has an endless amount of distinct objects. For example, the set of all natural numbers {1, 2, 3, 4, ...} is an infinite set.

What is a limit point?

A limit point of a set is a point where every neighborhood of that point contains at least one point from the set. In other words, if a point is a limit point of a set, it means that the point is "approached" by the set from all directions.

What is the difference between an accumulation point and a limit point?

An accumulation point is a point where every neighborhood of that point contains infinitely many points from the set. This means that the points are "clustered" around the accumulation point. A limit point, on the other hand, only requires that every neighborhood of the point contains at least one point from the set.

How are finite sets and limit points related?

Finite sets and limit points are related in that a finite set can have limit points, but not all finite sets have limit points. For example, the set {1, 2, 3} has no limit points, but the set {1, 1/2, 1/3, ...} has a limit point of 0. However, infinite sets are more likely to have limit points due to their endless nature.

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