# Irregularity of almost sure convergence

Let $y_n$ be a sequence on topological space. If $\forall$ subsequence $y_{n_k}$, $\exists$ a further subsequence $y_{n(m_k)}$ s.t. $y_{n(m_k)} \to y$, then $y_n \to y$.
A sequence of random variables $X_n \to X$ in probability $\iff$ $\forall$ subsequence $X_{n_k}$, $\exists$ a further subsequence $X_{n(m_k)}$ s.t. $X_{n(m_k)} \to X$ a.s.

Theorem 1 and 2 combined implies that almost sure convergence does not come from topology. In fact, while convergence in probability forms convergence class, a.s. convergence does not. This shows that a.s. convergense is actually not a “convergence” concept that we generally think of.