Let $V: = V_0 \supseteq V_1 \supseteq V_2 \supseteq V_3 \supseteq \ldots$ be a descending filtration of a vector space $V$. Then one can define as basis of open sets given by $\{ v + V_k, v \in V, k \in \mathbb{N}\}$ generating a topology on $V$. The equivalence classes of Cauchy sequences in this topology is defined to the completion of $V$. Equivalently it can also be defined as an inverse limit.
Is there an equivalent way to define a completion of a vector space/ring etc using ascending filtration? If $V_0' \subseteq V_1' \subseteq V_2' \subseteq \ldots$ is an ascending filtration, can one define a basis $\{ v + V_k', v \in V, k \in \mathbb{N}\}$ ? How does this completion look like?
In this topology, a set is open iff for every vector $v$ it also contains $v + V'_0$. So 1. the ascending filtration is not used in any way (except for the first subspace in it), 2. it is the same topology as for the constant descending filtration $V'_0 \supseteq V'_0 \supseteq V'_0 \supseteq \cdots$ and 3. the completion with respect to this topology is $V/V'_0$, so the topology is complete iff $V'_0=0$ iff the topology is trivial.