The concept of adaptive eigenspace basis (AEB) has recently proved effective for solving medium imaging problems. In this article, we present an AEB strategy for design parameterization in topology optimization (TO) problems. We seek the density design field as a linear combination of eigenfunctions, computed for an elliptic operator defined over the structural domain, and solve for the associated eigenfunction coefficients. Restriction to this truncated eigenspace drastically reduces the design dimension and imposes implicit regularization upon the solution, removing the need for auxiliary filtering operations and design-variable bound constraints. We furthermore develop the basis adaptation scheme inherent in the AEB, which iteratively recomputes the eigenfunction basis to conform to the evolving density field, enabling further dimension reduction and acceleration of the optimization process. The known aptitude of the adapted eigenfunctions to approximate piecewise constant fields is especially useful for TO as relevant design subspaces can be given low-dimensional representations. We propose criteria for the selection of the basis dimension and demonstrate the use of basis function selection as means for length scale control. We compare performance of the AEB against conventional TO implementations in problems for static linear-elasticity, showing comparable structural solutions, computational cost benefits, and consistent design dimension reduction.