- #1

dalcde

- 166

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Is it true that finite sets don't have limit points?

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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

NeroKid

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it depends on what topology u use on the unversal set

- #3

dalcde

- 166

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The real numbers and the Euclidean metric.

- #4

HallsofIvy

Science Advisor

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- #5

NeroKid

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- #6

henry_m

- 160

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But it's true for every T

- #7

dalcde

- 166

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Thanks!

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.

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.

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.

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.

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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