Proof That Every Open Set in a T1 Space Containing an Accumulation Point Contains an Infinite Number of Points

Theorem

Letbe a T1 space. Ifis an accumulation point of a subsetofthen every open set containing a contains an infinite number of points of

Proof

Supposeis an accumulation point ofand supposeis an open subset of withand that

is a finite subset of a T1 space, so closed, andis open.

Letthenis open,anddoes not contain any points of different fromHenceis 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.