A topological $n$-partition of a Hausdorff space $(X,\mathcal{O})$ is an $n$-partition $\Pi\in P(X)$ (in the sense of Hartmanis) which satisfies an additional topological condition. A subbase for the topology $\mathcal{O}_\Pi$ of $\Pi$ consists of all sets $O^*=\{p\in\Pi\mid p\cap O\neq\emptyset\}$ where $O\in\mathcal{O}$. The central question in the paper is: if $\mathcal{P}$ is a topological property and $X$ has $\mathcal{P}$, does $\Pi$ have $\mathcal{P}$? Preserving of topological properties is investigated for an arbitrary $X$ and in the special situations when $X$ is a locally compact or a compact space, or $\Pi=[X]^n$. If $X$ is compact or $\Pi=[X]^n$, then $\Pi$ is a subspace of $2^X$ with the Vietoris topology. Topological projective and Euclidean planes are the special topological 2-partitions.