For questions about complex manifolds.
A topological manifold of dimension $n$ is a Hausdorff, second countable topological space $X$ such that each $x \in X$ has a neighbourhood homeomorphic to an open subset of $\mathbb{R}^n$.
A complex atlas on a $2n$-dimensional topological manifold $X$ is a collection of pairs $\{(U_{\alpha}, \varphi_{\alpha})\}_{\alpha\in A}$ where $U_{\alpha}$ is an open subset of $X$ and $\varphi_{\alpha} : U \to \varphi_{\alpha}(U)$ is a homeomorphism, where $\varphi_{\alpha}(U_{\alpha})$ is the open disc in $\mathbb{C}^n$, such that $\bigcup\limits_{\alpha\in A}U_{\alpha} = X$.
A topological manifold with a maximal complex atlas is called a complex manifold.
