CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A subset $S$ of a real affine space $X$ is convex if for any two points $x,y \in S$, the straight line segment connecting $x$ with $y$ in $X$ is also contained in $S$. In other words, for any $x,y\in S$, and any $t\in [0,1]$, we have also $t x + (1-t) y \in S$.
Every convex set is star-shaped.
One generalization of convexity to Riemannian manifolds and metric spaces is geodesic convexity.
An abstract generalization of the notion of a convex set is that of a convex space. Note that as mentioned there, there is a nice characterization of those convex spaces which are isomorphic to convex subsets of real affine space.
The convex hull of a subset is the smallest convex subset containing it.