Why are these 'counter' examples in topology?

Question

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?

Answer 1

Metric spaces are 1st countable.
Compact metric spaces are 2nd countable.

Answer 2

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.