Examples about compactness Compactness implies countably compactness which in turn implies limit-point compactness. 
Sequentially compactness implies limit point compactness. 
$Z_{+} \times \{0,1\}$ with two-point indiscrete space $\{0,1\}$ is limit-point compact but not sequentially compact. 
First uncountable ordinal is sequentially compact, countably compact, but not compact.   
Can anybody give me any examples for 


*

*Sequentially compact but not countably compact space (hence limit-point compact but not compact)

*Countably compact but neither compact nor sequentially compact space.
?
Thanks in advance.

Actually, sequentially compact implies countably compact, so no example for (1) exist. The proof is actually quite straightforward:

Assume that $X$ is not countably compact. Then there exists a countable open cover $X = \bigcup_{n=1}^\infty U_n$ that has no finite subcover.
For each $m$, let $x_m \in X \setminus \bigcup_{n=1}^m U_n$. The sequence $x_m$ cannot have a limit point $x \in X$, otherwise it would be true that $x \in X \setminus (\bigcup_{n=1}^m U_n)$ for every fixed $m$, which implies that $x \in X \setminus (\bigcup_{n=1}^\infty U_n) = \emptyset$. Therefore, the sequence $x_m$ has no convergent subsequence.

As for (2), there's an example in the book Counterexamples in Topology, by  Steen and Seebach (example 106): Let $\omega_1$ be the smallest uncountable ordinal with the order topology, and let $I = [0,1]$. Then $X = \omega_1 \times I^I$ is countably compact (because $\omega_1$ is sequentially compact and $I^I$ is compact) but neither sequentially compact nor compact (these properties are preserved by the projections).
