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 about each of its points, and hence contractible provided it is inhabited.
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.