CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
An open cover of a topological space $X$ is a collection $\{U_i \subset X\}$ of open subsets of $X$ whose union equals $X$: $\cup_i U_i = X$.
When denoting by $U_i \hookrightarrow X$ the inclusion morphisms in the category Top, each open cover constitutes a covering family $\{U_i \to X\}$ in the sense of sheaf and topos theory which is a standard coverage on Top.
Analogous statements hold for categories of topological spaces with extra structure, such as the category Diff of smooth manifolds.
If an open cover has the property that all the $U_i$ and all of their finite nonempty intersections are contractible, then one speaks of a good open cover.