Abstract: The Johnson-Lindenstrauss lemma says that you can project sets points in a high dimensional space to a much smaller dimension that roughly preserves pairwise distances. In fact, with some probability, a random projection will suffice. We prove a similar result holds when considering topological information. In particular, with some probability, random projections to a lower dimension does not significantly change the homology of a union of balls in Euclidean space. This leads to a practical probabilistic algorithm for calculating persistent homology for high dimensional spaces.