Are the following definitions of relative countable compactness equivalent?
$\overline{A}$ is countably compact
Every sequence in $A$ has a convergent subnet
Note that for regular spaces, the analoguous definitions are equivalent for relative compactness. Also, the equivalence always holds if $A$ is closed.
Edit: Note that if $A\subseteq X$, then taking $A = X$ we obtain two equivalent definitions for countable compactness. This begs the question if they are equivalent for, this time relative, countable compactness, if we assume that $X$ is a nice enough topological space. If there are two different ways to expand on a concept then we'd like to know if those ways are equivalent for nice enough examples. We could do something similar for compactness, and it turns out the definitions are equivalent if $X$ is regular, so one can hope for something similar for countable compactness.
These are not equivalent, even for completely regular spaces. For a counterexample, let $X$ be the deleted Tychonoff plank $(\omega_1+1)\times(\omega+1)\setminus\{(\omega_1,\omega)\}$ and let $A=\omega_1\times(\omega+1)\subset X$. Then $A$ satisfies (2), and in fact is countably compact itself (since every sequence in $A$ is bounded in the first coordinate). However, $\overline{A}=X$ is not countably compact, since the sequence $(\omega_1,n)$ has no accumulation point in $X$.