When almost all points are not isolated in all subspaces

Question

Let $X$ be a compact (non Hausdorff) $T_0$ topological space such that for any subset $\mathcal{A}=\{\mathfrak{x}_\alpha\}_{\alpha\in \Lambda}$ of distinct element of $X$ the set $\{\mathfrak{x}_\beta\}$ is not an open subset of $\mathcal{A}$ for all but finitely many $\beta\in \Lambda$, when considering $\mathcal{A}$ as a subspace of $X$. Is there any description or characterization for such a space?

Answer 1

First, $\mathbf{X}$ having a subspace with infinitely many isolated points is just equivalent to $\mathbb{N}$ embedding into $\mathbf{X}$. So we are looking at the topological spaces that $\mathbb{N}$ does not embed into.

We can get a "positive" characterization of these as follows:

Proposition: The following are equivalent:

1. $\mathbb{N}$ does not embed into $\mathbf{X}$
2. For any sequence $(U_n)_{n \in \mathbb{N}}$ of open sets there is some $k \in \mathbb{N}$ with $U_k \subseteq \bigcup_{n \in \mathbb{N} \setminus \{k\}} U_n$

Proof: If $\mathbb{N}$ embeds into $\mathbf{X}$, let $U_n$ be the open set in $\mathbf{X}$ that ensures that $\{n\}$ is open in the embedded subspace. Then $(U_n)_{n \in \mathbb{N}}$ clearly violates Property (2).

Conversely, if you have a violation $(U_n)_{n \in \mathbb{N}}$ of property (2), we can chose some $x_n \in U_n \setminus \left (\bigcup_{\ell \in \mathbb{N} \setminus \{n\}} U_\ell \right )$, and find that $n \mapsto x_n : \mathbb{N} \to \mathbf{X}$ is an embedding.