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.
Continuous Mapping and Connectedness
compactness-containing-union-of-subsets-lemma-bolzano-weierstrass-theorem-subsequence