Wikipedia's definition of a topological field is as follows
Let $(\mathsf{F}, \mathcal{T}_\mathsf{F})$ be a topological space and $(\mathsf{F}, +, \times)$ be a field. We say that $(\mathsf{F}, \mathcal{T}_\mathsf{F}, +, \times)$ is a topological field if addition, multiplication, $i_+:a\mapsto -a$ and $i_\times:a\mapsto a^{-1}$ are continuous maps.
Issue: Continuity of $a\mapsto a^{-1}$
I have an issue understanding continuity of the map $i_\times$. This map is defined on all of $\mathsf{F}$ except the additive identity $0_\mathsf{F}$ $$ i_\times:\mathsf{F}\backslash\{0_\mathsf{F}\}\to\mathsf{F}\backslash\{0_\mathsf{F}\}. $$ Continuity would then mean that for any $\mathsf{A}\in\mathcal{T}_{\mathsf{F}\backslash\{0_\mathsf{F}\}}$ one has $i_\times^{-1}(\mathsf{A})\in\mathcal{T}_{\mathsf{F}\backslash\{0_\mathsf{F}\}}$. How do I know that I can find a topology for $\mathsf{F}\backslash\{0_\mathsf{F}\}$? How do I know $(\mathsf{F}\backslash\{0_\mathsf{F}\}, \mathcal{T}_{\mathsf{F}\backslash\{0_\mathsf{F}\}})$ is a topological space?