Theorem
Letbe 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
Supposeis 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.
Letthen
is open,
and
does not contain any points of
different from
Hence
is 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.