In the discussion on convergence in topological spaces, the main focus is on proving that a sequence \( x_n \) converges to \( x \) if and only if the distance \( d(x_n, x) \) approaches zero as \( n \) approaches infinity. Participants clarify whether \( d \) represents a metric, noting that the problem may not apply to non-metrizable spaces. The conversation emphasizes the importance of defining "limit" in the context of metric spaces. The standard definition of convergence in metric spaces is referenced as a potential framework for the proof. Overall, the discussion seeks to establish a clear understanding of convergence within the specified topological context.