  Datasets are mathematical objects (e.g., point clouds, matrices, graphs,
images, fields/functions) that have shape. This shape encodes important
knowledge about the system under study. Topology is an area of mathematics that
provides diverse tools to characterize the shape of data objects. In this work,
we study a specific tool known as the Euler characteristic (EC). The EC is a
general, low-dimensional, and interpretable descriptor of topological spaces
defined by data objects. We revise the mathematical foundations of the EC and
highlight its connections with statistics, linear algebra, field theory, and
graph theory. We discuss advantages offered by the use of the EC in the
characterization of complex datasets; to do so, we illustrate its use in
different applications of interest in chemical engineering such as process
monitoring, flow cytometry, and microscopy. We show that the EC provides a
descriptor that effectively reduces complex datasets and that this reduction
facilitates tasks such as visualization, regression, classification, and
clustering.
