The exploration of open graphs reveals insights into Tukey order within topological spaces, implying stronger convergence properties.
The two alternatives G ≤ K ℵ 0 G≤ K_0 and K ℵ 1 ≤ G K_1≤ G of the Open Graph Dichotomy are explored here for a natural class of graphs associated to points x x in topological spaces X X . This is then used in finding Tukey restrictions on the neighbourhood filters of points x x guaranteeing that they have strong properties of sequential convergence. This paper can be seen as natural continuation of Todorcevic [Fund. Math. 267 (2024), pp. 181–194] that gives a Tukey theory interpretations of alternatives of another combinatorial dichotomy.
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Stevo Todorčević (2025) studied this question.
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