Dimensionality reduction (DR) methods are often used to unveil important information from a dataset. Most advanced DR algorithms are non-linear, presenting better performance in the result at the cost of a lower interpretability of whether the reduced dataset resembles the original one. While new methods for dimensionality reduction arise, UMAP is still more used than any other, for its great performance and solid theoretical foundations. As an embedding operation, DR methods induce a loss of information. Evaluation measures can be used in order to quantify the (local) faithfulness of the reduction. In this paper we propose using fuzzy simplicial sets (the cornerstone of UMAP) to develop evaluation measures. We demonstrate the usefulness of these measures in detecting unfaithfulness by revealing structural distortions, and identifying when the reduction falls below the intrinsic dimensionality.