Over the summer I looked at the connections between discrete Morse theory and finite topological spaces. Discrete Morse theory lets you analyze the topology of discrete spaces using a special kind of function. Discrete Morse theory has applications in areas like combinatorics, topological data analysis, and computer science. A finite topological space is a space that contains a finite number of points, which can surprisingly have some interesting topology. Using the ideas of discrete Morse theory, we can study some fintie topological spaces using the same tools.