Estimation of the number of sources in a linear mixture is a critical preprocessing step in the separation and analysis of the sources for many applications. Historically, statistical methods, such as the minimum description length and Akaike information criterion, have been used to estimate the number of sources based on the autocorrelation matrix of the received mixture. In this paper, we introduce an alternative, topology-based method to compute the number of source signals present in a linear mixture for the class of constant-amplitude, monocomponent source signals. As a proof-of-concept, we include an example of three such source signals that overlap at multiple points in time and frequency, which the method correctly identifies from a set of eight redundant measurements. These preliminary results are promising and encourage further investigation into applications of topological data analysis to signal processing problems.