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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.

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  • $\begingroup$ I deleted my answer because it didn't really answer your question and it fits in a comment: the surreal numbers under the topology given on page 5-6 of this paper (arxiv.org/pdf/1307.7392.pdf) by Rubinstein-Salzedo and Swaminathan are a non-metric continuum satisfying definitions (2) and (3), however the authors offer a slightly altered definition of strong compactness on 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
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A good starting point for studying countinua in the sense of the definition (2), i.e., compact Hausdorff spaces, is the survey paper "Continuum Theory (General)" by Ed Tymchatyn in Encyclopedia of General Topology. This paper contains further references.

As an example of a non-trivial result holding for both definitions (1) and (2) one can mention that the superextension of a compact Hausdorff space is connected and locally connected. This can be found in the old book of van Mill.

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Here is a non-trivial property commonly known for definition (1) that can be shown to also hold for definitions (2) or even (3):

If we have a family of continua $(X_n)_{n\in\kappa}$ for $\kappa$ being an infinite cardinal with the property that for $n \leq m\in \kappa$ we have $X_n \supseteq X_m$, then $\bigcap_{n\in\kappa} X_n$ is a continuum.

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The non-cut point existence theorem is a good one.

Wilder proved sometime around the 1930's that every type (1) continuum has at least two non-cut points. This was later generalized for type (2) continua sometime around the 1960's by Whyburn, and finally, for all (type (3)) continua in 1999 by Honari and Brahmanpour.

Most results do not proceed this way, I must warn you.

Usually either (a) there is a non-metric counterexample, or (b) the original proof for metric continua "obviously" does not require metrizability and so applies to more general continua.

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