Corollary 4

Let E be a vector space over a topological division ring K, and let $(\mathscr{T}_i)_{i \in I}$ be a family of topologies compatible with the vector space structure of E; then the upper bound $\mathscr{T}$ of the topologies $\mathscr{T}_i$ is compatible with the vector space structure of E.

For, if $E_t$ denotes the topological vector space obtained from $E$ by the topology $\mathscr{T}_t$, and $f_t$ the identity map of $E$ on $E_t$, then $\mathscr{T}$ is the coarsest topology making the $f_t$ continuous.