In class the teacher gave us the definition of a continuum as a metric, connected and compact space, on the book General Topology by Willard says that is a connected, compact and Hausdorff space, and in here only says that is a compact and connected space.
I'm confused, which is it? I can imagine how the first two could relate, since metric implies Hausdorff, and maybe they are even equivalent, I have to be honest I haven't checked that, but what about the third? It's way more general, is it wrong?
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$\begingroup$Usually a continuum means either
(1) connected compact Hausdorff, or
(2) connected compact metric.
Authors should make it clear which definition they are using. In my papers, I use the first definition, then if I use "metric continuum" to describe a space satisfying (2). The theory of non-metric continua is very interesting, so that we should not limit ourselves to metric continua. Note that many fundamental theorems in continuum theory hold for non-metric spaces, while others require metrizability (which is equivalent to second countability in compact Hausdorff spaces). For instance, the Boundary Bumping Theorem holds in non-metric spaces, but to prove that an indecomposable continuum has more than one composant we must assume the continuum is metric.
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