Dimension reduction is one of the most important issues in machine learning and computational intelligence. Typical data sets are point clouds in a high dimensional space with a hidden structure to be found in low dimensional submanifolds. Finding this intrinsic manifold structure is very important in the understanding of the data and for reducing computational complexity. In this paper, we propose a novel approach for dimension estimation of topological manifolds based on measures of simplices. We also investigate the effects of resolution changes for dimension estimation in the framework of Morse theory. The result is a method that can be used for data located in simplical complexes of varying dimensions and with no continuous or differentiable structure. The proposed method is applied to images of handwritten digits with known deforming dimensions, data with a nontrivial topology and noisy data. We compare the estimates with results obtain by local PCA.