Proof That Every Open Set in a T1 Space Containing an Accumulation Point Contains an Infinite Number of Points
Theorem
Let
be a T1 space. If
is an accumulation point of a subset
of
then every open set containing a contains an infinite number of points of
Proof
Suppose
is an accumulation point of
and suppose
is an open subset of
with
and that
is a finite subset of a T1 space, so closed, and
is open.
Let
then
is open,
anddoes not contain any points of different from
Henceis not an accumulation point of
The converse - that every open set containing a contains some point ofdifferent from- is true by definition of accumulation point.
