What important results hold for non-metric continua, or where can I find a survey of such results?

There are three definitions of a continuum around: a non-empty topological space that is

(1) connected compact metric, or

(2) connected compact Hausdorff [e.g., *General Topology* by Willard], or

(3) connected compact [ProofWiki].

I am interested in non-trivial properties commonly known for definition (1) that have been found to also hold for definitions (2) or even (3).

I have asked this question on math.stackexchange but did not get any answer.

strong compactnesson page 25. Perhaps you could glean some useful information by seeing what properties are carried over from smaller real-closed continuua, all the way down to $\mathbb{R}$. $\endgroup$ – Alec Rhea Nov 5 '17 at 7:47