Topology counterexamples without ordinals

I am looking for three counterexamples in general topology, namely:

• A set which is sequentially closed, but not closed;
• A set which is sequentially compact, but not compact;
• A set which is compact but not sequentially compact.

I do know some examples of such spaces but they all involve ordinals, which I am trying to avoid. Is it possible to construct some counterexamples without them?

It's too bad these classics are not better-known:

Let $Y=[0,1]^{\Bbb R}$ in the product topology.

Then $Y$ is compact, but not sequentially compact.

Let $X=\{f \in Y: \{t \in \Bbb R: f(t) \neq 0\} \text{ is at most countable} \}$.

Then $X$ is sequentially closed in $Y$ (but not closed in $Y$, it's even dense).

$X$ is sequentially compact (and pseudocompact and countably compact too) in the subspace topology inherited from $Y$ but not compact.

Instead of $\Bbb R$ we could use almost any uncountable set (except for the first where taking an index set of size continuum is the safe choice: it's consistent with ZFC that $[0,1]^{\aleph_1}$ is sequentially compact too). I chose the reals for definiteness and familiarity.