A [[Vector spaces]], $\mathcal{E}$, is topological over a [[topological field]] $\mathbb{F}$ if its topological structure and linear structure are compatible such that, 1) Subtraction (or equivalently addition), $\mathcal{E}\times\mathcal{E}\rightarrow\mathcal{E}$ is [Continuous](Continuity.md) 2) And the multiplication of scalars $\mathbb{F}\times \mathcal{E} \rightarrow \mathcal{E}$ is [Continuous](Continuity.md). When referring to $\mathbb{F}$ assume (unless stated otherwise) that we're dealing with $\mathbb{R}$ or $\mathbb{C}$. In addition to the above two conditions, some sources require $\mathcal{E}$ to be [Hausdorff](Hausdorff%20separation%20axiom.md). %%Why make this assumption? I don't like this%% #MathematicalFoundations/Geometry/Topology #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces