compactness / sequentially compact

Question

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

Answer 1

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

Answer 2

Example 1 with proof: Stone-Čech Compactification of the Integers $\beta \omega$

Proof: It is compact obviously. We will prove that $\beta\omega$ is not sequentially compact. Note that every infinite set in $\beta\omega$ has $2^\mathfrak c$ cluster points, hence the only convergent sequences in $\beta\omega$ are those which are eventually constant; therefore if $X$ is a subspace of $\beta\omega$ and $X$ is sequentially compact, then $X$ is finite. So $\beta\omega$ cannot be sequentially compact.