Topological Data Analysis (TDA) focuses on the inherent shape of (spatial) data. As such, it may provide useful methods to explore spatial representations of linguistic data (embeddings) which have become central in NLP. In this paper we aim to introduce TDA to researchers in language technology. We use TDA to represent document structure as so-called story trees. Story trees are hierarchical representations created from semantic vector representations of sentences via persistent homology. They can be used to identify and clearly visualize prominent components of a story line. We showcase their potential by using story trees to create extractive summaries for news stories.