compactness / sequentially compact

general-topology compactness examples-counterexamples

I'm looking for two examples:

  1. A space which is compact but not sequentially compact
  2. A space which is sequentially compact but not compact

Explanations why the spaces are compact / not compact and sequentially compact / not sequentially compact would be appreciated. A reference would also be appreciated. So the conclusion would be, that there's no equivalence in general. Of course they are equivalent in a metric space.

math


The following examples are from $\pi$-Base, a searchable database of Steen and Seebach's Counterexamples in Topology.

(Click on the following links to learn more about the spaces.)

For compact but not sequentially compact:

  • Stone-Cech Compactification of the Integers
  • Uncountable Cartesian Product of Unit Interval ($I^I$)

For sequentially compact but not compact:

  • An Altered Long Line
  • $[0, \omega_1)$ ($\omega_1$ is the first uncountable ordinal)
  • The Long Line
  • Tychonoff Corkscrew

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