Topological point states (TPSs) have the same physical mechanism as that of the topological corner states (TCSs) in higher-order topological insulators (HOTIs) in two-dimensional (2D) systems and bring a new physical perspective and application prospect to HOTI. However, the current research on TPSs is limited to periodic structures and Hermitian systems. The special structure of quasicrystals and the skin effect in non-Hermitian systems may bring new physical phenomena to TPSs. In this paper, the TPSs are discovered in non-Hermitian quasicrystals, and the consistency between the two point vacancies for TPSs and the two ends of a 1D topological chain is demonstrated. TPSs will provide a method for the localization control of non-Hermitian quasicrystals.

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• Symmetry protected topological states
• Topological insulators
• Topological phase transition
• Topological phases of matter
• Non-Hermitian systems
• Quasicrystals