CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A cover (open cover) $\{U_\alpha\}$ of a topological space $X$ is locally finite if for all $x\in X$, there is an neighbourhood $N \ni x$ such that $N \cap U_\alpha \neq \emptyset$ for only a finite number of $\alpha$.
Open covers defined by partitions of unity are locally finite. (CHECK!)