Generalizing the T_0 separation axiom

The starting point of this blog post is a slight reformulation of the T_0 separation axiom: A topological space (X,\tau) is T_0 if for all x\neq y\in X there is a set U\in \tau such that

\{x,y\}\cap U \neq \emptyset \text{ and } \{x,y\}\not\subseteq U.

Given a cardinal \kappa \geq 2, we say that a space (X,\tau) is T^{\kappa}_0 if for all subsets S\subseteq X with |S|=\kappa there is a set U\in \tau such that U “splits” S, or more formally

S\cap U \neq \emptyset \text{ and } S\not\subseteq U.

Obviously, if \lambda\geq \kappa\geq 2 and if (X,\tau) is T^\kappa_0, then X is also T^\lambda_0. We say, the space (X,\tau) is minimally T^\kappa_0 if it is T^\kappa_0, but for all cardinals \alpha<\kappa with \alpha\geq 2, the space (X,\tau) is not T^\alpha_0.

Question. Given cardinals \lambda\geq\kappa\geq 2, is there a topological space (X,\tau) such that |X|=\lambda and (X,\tau) is minimally T^\kappa_0?

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About dominiczypen

I'm interested in general topology, order theory, and graph theory. This link takes you to my preprints on arXiv.
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