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In Counterexamples in Topology from Steen & Seebach I found the following compact and Hausdorff counterexamples with some properties:

- Lexicographically ordered square: 1st-countable, not separable, not 2nd-countable, not metrizable, connected, not path-connected.
- Concentric circles: 1st-countable, completely normal, not separable, not 2nd-countable, not metrizable.
- Helly Space: Separable, 1st-countable, not 2nd-countable, not metrizable, sequentially compact.
- Double Arrow: Separable, 1st-countable, not 2nd-countable, homogeneous, not metrizable.

I am trying to see exactly why these spaces behave differently from compact metric spaces, but because of a lack of intuition in the properties of compact metric spaces, I can not see why these topological examples are special.
Could you please give me some theorems/properties which compact metric spaces do behave like, but these examples don't?

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  • $\begingroup$ The listed spaces are compact. Hence there is no "theorem/properties which compact spaces do behave like, but these examples don't". However, a common intuition is that compact spaces are almost as good as finite sets. Then properties like not separable might by counter-intuitive ... $\endgroup$ – Hagen von Eitzen Jun 29 '17 at 12:53
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    $\begingroup$ @HagenvonEitzen Thanks for your comment. You forgot the word 'metric' in the quote though. I am searching for theorems/properties which metric compact spaces do behave like, but these topological compact spaces do not behave like. Being not separable is indeed a good example, thank you! $\endgroup$ – Math Learner Jun 29 '17 at 13:02
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Metric spaces are 1st countable.
Compact metric spaces are 2nd countable.

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  1. All metric spaces are $T_6.$ The lex-order-topology on $[0,1]^2$ is $T_5,$ as are all linear spaces, but this one is not $T_6$: Singleton subsets are $G_{\delta}$ but the closed set $[0,1]\times \{0,1\}$ is not a $G_{\delta}$ set.

  2. Separable metric spaces are 2nd-countable. Compact metric spaces are separable. The Helly Space and the Double-Arrow space are separable but not 2nd-countable.

  3. I may have seen the Two Circles Space but by a different name.

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