Do Finite Sets Really Lack Limit Points?

[dalcde]

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

 

[NeroKid]

it depends on what topology u use on the unversal set

 

[dalcde]

The real numbers and the Euclidean metric.

 

[HallsofIvy]

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 $\delta> 0$, there exist a point, q, of A other than p such that $d(p,q)< \delta$. 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 $\delta$ to be smaller than M shows that A cannot have any limit points.

 

[NeroKid]

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

 

[henry_m]

Not true in general for topological spaces. For example, for any set X with the indiscrete topology $\{\emptyset,X\}$, 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.

 

[dalcde]

Thanks!