# Example for Sequentially Compact space which is not Compact not involving ordinals?

In all the counterexamples to the assertion $$\text{Sequentially Compact} \implies \text{Compact}$$, they involve ordinals in some form of the other, such as

1. Ordinal Space
2. Long Ray/Line
3. Altered Long Ray/Line

to name the ones I could find.

Since I have no knowledge of ordinals, I was wondering if there were counterexamples to the above assertion which does not involve ordinals. Also, if there does not exist such counterexamples, is there a deeper reason for this? Or is it just that such examples are not known?

• Try $$X = \Bigl\{ f \in \{0,1\}^{\mathbb{R}} : \{t \in \mathbb{R} : f(t) \neq 0\} \text{ is countable}\Bigr\}\,.$$ It's dense in $K = \{0,1\}^{\mathbb{R}}$, hence not compact. But every sequence in $X$ is contained in a compact metrisable subset of $K$, hence has a convergent subsequence. Jul 27, 2020 at 19:58
• @Ishan Part of the reason that these counterexamples look a little weird and arbitrary is that for metric spaces, "compact" and "sequentially compact" are equivalent, which rules out many typical spaces (including subsets of $\Bbb{R}^n$ in the subspace topology) from serving as counterexamples. Jul 27, 2020 at 23:06

The $$\Sigma$$-product is another good one (also suggested by Daniel in the comment)
$$X=\{f \in \{0,1\}^{\Bbb R}: \operatorname{supp}(f):=\{x: f(x) \neq 0\} \text{ is at most countable }\}$$
where in fact any uncountable index set (as $$\Bbb R$$, as we chose) will do.
This is countably compact, sequentially compact, pseudocompact but not first countable nor compact, and a topological group as well (under the mod 2 pointwise addition, so a subgroup of $$\{0,1\}^{\Bbb R}$$). If $$\{f_n: n \in \Bbb N\}$$ is any countable subset of $$X$$, then the "total support" $$A:= \bigcup_n \operatorname{supp}(f_n)$$ is also at most countable and $$\{0,1\}^A$$ is a compact metric space in which this set essentially lives. A limit point, or subsequence limit there can then be found, stuffed with $$0$$'s on the other coordinates and we get one for the sequence/set we started with in $$X$$. Non-compactness is obvious as $$X$$ is dense in $$\{0,1\}^{\Bbb R}$$, making it also a ccc (but non-separable) space as well. It's a very nice example in many ways.