Proposition: A set is connected if and only if it is an interval, that is, of the form [a,b], [a,b), or (a,b) where a or b can be plus or minus infinity.
Proposition: If A in n-dimensional R space is open and connected, then it is path-connected.
compactness is a collection of sets
compactness in terms of a collection of sets and subsets. converges to a point in the domain of the metric space.