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limit point
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Mon Jun 17 2002 at 5:35:03
(
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 space
s.
printable version
chaos
accumulation point
closed
closed set
limit
limit break
set theory
Topological Space
An interesting application of binary search
Heine-Borel Theorem
topology
metric space
Closure
compact
derived set
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