Examples of sequential compact but not compact spaces that do not use ordinals.

Question

I think the title is self explanatory, I'm using Munkres' Second Edition text for Point Set Topology and I can't figure out if such examples are possible.

Answer 1

Let $P=\{0,1\}^\mathbb{R}$ in the product topology.

Define the subspace $X = \{f \in P: f^{-1}[\{1\}] \text{ is at most countable }\}$ in the subspace topology, all functions that have are almost always $0$.

$X$ is dense in $P$ (which is compact Hausdorff, but not sequentially compact), so not compact (or it would have been closed, not dense), so $X$ is completely regular.

$X$ is countably compact and sequentially compact: this follows from the fact that any countable subset of $X$ (or sequence from $X$) "essentially lives" on a product $\{0,1\}^M$ with $M$ countable (which is compact metric and hence sequentially compact too).

$X$ is also a sequentially closed subset of $P$ that is not closed. It's also a topological group. It's quite a nice example and requires no more knowledge than countability and the uncountability of $\mathbb{R}$.