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Grenzpunkte

Grenzpunkte is a term used in German mathematics that can refer to different concepts depending on the context. In many texts it denotes points related to the boundary of a set, while in others it is used for limit points (accumulation points) of a subset in a topological space.

One meaning is the limit point (Häufungspunkt) of a subset A of a topological or metric space

Another meaning is the boundary point (Randpunkt) of a set A. A point x ∈ X is a

Relation and examples help clarify distinctions. For A = (0,1) in R with the usual metric, the Grenzpunkte

In summary, Grenzpunkte can denote either limit points or boundary points of a set, depending on the

X.
A
point
x
∈
X
is
a
Grenzpunkt
of
A
if
every
neighborhood
U
of
x
contains
a
point
of
A
different
from
x.
The
collection
of
all
Grenzpunkte
of
A
is
called
the
derived
set
A'.
A
Grenzpunkt
may
lie
inside
A
or
outside
it;
a
point
can
be
a
Grenzpunkt
even
if
it
belongs
to
A.
Randpunkt
of
A
if
every
neighborhood
of
x
intersects
both
A
and
the
complement
X
\
A.
The
set
of
all
Randpunkte
is
the
boundary
∂A,
which
can
also
be
described
as
∂A
=
closure(A)
∩
closure(X
\
A)
or
as
∂A
=
closure(A)
\
interior(A).
Some
texts
refer
to
Randpunkte
as
Grenzpunkte
as
well,
leading
to
potential
ambiguity
between
the
two
concepts.
(limit
points)
of
A
are
0
and
1;
these
points
are
also
the
boundary
points.
In
a
set
with
an
isolated
boundary
point,
a
boundary
point
can
fail
to
be
a
limit
point,
illustrating
that
the
two
notions
are
related
but
not
identical.
mathematical
context.