Accumulation point
is an accumulation point of when , so has points arbitrarily close to other than itself.
Definition 2.1 (Accumulation Point)
Let , and . We say that is an accumulation point for if
If and is not an accumulation point for , we say that is an isolated point of .
Sequential characterisation
A point is an accumulation point of if and only if there exists a sequence in with as : each threshold supplies some with , and the thresholds build such a sequence; conversely every term of a sequence converging to lies within any given of for large.
Examples
-
For , the point is isolated while every point of is an accumulation point of .
-
Every point of the circle is an accumulation point of the disc .
-
All points of are accumulation points of , and the same holds for the closed set .
Related
Stated in
- Definition 2.1 (Accumulation Point)§2.1 Limits of Functions