(
Mathematics -
Set Theory and Topology)
Definition of limit point
Let
x be a point and
S be a
subset of a
metric space M. Then
x is a
limit point of
S if the
closure of
S - {
x} includes
x.
Definition of closure
For any subset
A in
M,
A' is the
closure of
A if it is the intersection of every
closed subset of
M that contains
A. In other words,
A' is the smallest closed set
in
M that contains
A.
The above definitions come from
Set Theory and Metric Spaces by Kaplansky © 1972. A different definition of limit point comes from
The Advanced Calculus of One Variable by Lick © 1971. Lick's version is as follows:
Definition of limit point
Let
S be a set of
real numbers. Point
x is a
limit point of
S if there exists infinitely many elements of S in each
neighborhood of
x.
In
Complex Variables and Applications 6th Ed. Brown & Churchill © 1996, the term
accumulation point is given a definition similar to Lick's definition of limit point except for the broader
complex numbers.
In
Eric Weisstein's World of Mathematics, limit point is defined similarly to Kaplansky's definition, except it applies to the broader
topological spaces.