compact space and limit point of a filter

1. Proposition

Let \((X, \mathcal{T})\) be a topological space. TFAE:

  1. \(X\) is compact, as in there exists a finite subcover
  2. each filter has a limit point

2. Proof

2.1. 1) \(\implies\) 2)

2.2. 2) \(\implies\) 1)

Date: nil

Author: Anton Zakrewski

Created: 2024-10-13 So 15:42