  It is well-known that the $T$-fixed points of a Schubert variety in the flag
variety $GL_n(\mathbb{C})/B$ can be characterized purely combinatorially in
terms of Bruhat order on the symmetric group $\mathfrak{S}_n$. In a recent
preprint, Cho, Hong, and Lee give a combinatorial description of the $T$-fixed
points of Hessenberg analogues of Schubert varieties (which we call Hessenberg
Schubert varieties) in a regular semisimple Hessenberg variety. This note gives
an interpretation of their result in terms of Bruhat order by making use of a
partition of the symmetric group defined using so-called subsets of Weyl type.
The Appendix, written by Michael Zeng, proves a lemma concerning subsets of
Weyl type which is required in our arguments.
