Generally speaking, we will have thre versions of every property a set might have: what I’ll call a topological version, involving open sets; a metric version, involving \(\epsilon\)s; and what I’ll call a sequential version, involving sequences. In any metric space (hence, in any normed space), these two notions will coincide. It’s more or less true that metric topology can be done entirely with sequences.
Proposition3.5.1.topological characterization of sequence limit.
Let \((V,\lVert\cdot\rVert)\) be a normed space and \((x_n)_{n\in \mathbb{N}}\) a sequence in \(V\text{.}\)\(x_n\to L\) if and only if for any open set \(U\) for which \(L\in U\text{,}\) there is a \(K\in \mathbb{N}\) so that \(n\gt K\) guarantees \(x_n\in U\text{.}\)
Proposition3.5.2.sequential characterization of closed.
Let \((V,\lVert\cdot\rVert)\) be a normed space and \(A\subseteq V\text{.}\) Then \(A\) is closed if and only for every convergent sequence \((a_n)_{n\in\mathbb{N}}\) with \(\forall n, a_n\in A\text{,}\) we have \(\lim a_n\in A\text{.}\)
Proposition3.5.3.sequential characterization of open.
Let \((V,\lVert\cdot\rVert)\) be a normed space and \(A\subseteq V\text{.}\) Then \(A\) is open if and only if for every point of \(a\) and every sequence \((x_n)_{n\in\mathbb{N}}\) with \(x_n\to a\text{,}\) there is \(K\) so that \(n\gt K\) guarantees \(x_n\in A\text{.}\)
