Bulk-boundary correspondence in non-Hermitian class- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">D</mml:mi> </mml:mrow> <mml:mo>†</mml:mo> </mml:msup> </mml:math> topological systems

The bulk-boundary correspondence in Hermitian systems is governed by wave-function topology, whereas in non-Hermitian systems exhibiting the skin effect, it is determined by complex energy topology. In this work, we establish the bulk-boundary correspondence in non-Hermitian class-${\mathrm{D}}^{\ifmmode\dagger\else\textdagger\fi{}}$ topological systems within the framework of wave-function topology. Specifically, we demonstrate that bulk polarization serves as a universal topological invariant, effectively characterizing anisotropic in-gap edge states in Hermitian class-D systems and isotropic in-gap edge states in non-Hermitian class-${\mathrm{D}}^{\ifmmode\dagger\else\textdagger\fi{}}$ systems. Additionally, the spectral winding number determines the localization direction of the isotropic in-gap edge states in class-${\mathrm{D}}^{\ifmmode\dagger\else\textdagger\fi{}}$ systems. By employing the Wilson loop approach to compute the bulk polarization, our work not only restores the bulk-boundary correspondence in non-Hermitian class-${\mathrm{D}}^{\ifmmode\dagger\else\textdagger\fi{}}$ systems, but it also provides valuable insights into the connection between wave-function topology and non-Hermitian band theory.