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This dissertation is on the theory of topological information in data and its applications. The last decade has witnessed a substantial development of topological data analysis (TDA) and its growing application in a broad spectrum of areas like biology, physics, astronomy, and data science. Rooted in both algebraic topology and metric geometry, TDA allows us to explore the underlying topology of the dataset by building approximating complexes upon the data points.In this dissertation, we focus on exploring lower dimensional topological features. We mainly consider two approaches to extract and analyze such lower dimensional topological features.One approach is to detect lower dimensional topological features from a (high-dimensional) dataset directly. In this direction, we investigate one specific type of topological feature consisting of the zero-density region of a density function from which the data points are drawn. We describe this feature of the density function, consider the asymptotic behavior of a sequence of coverings, and give practical guidelines for non-asymptotic situations.Another collection of approaches to extract lower dimensional features is less direct. We summarize a pair of papers that show how lower dimensional topological information can be used in the segmentation problem and the dimension reduction problem. The segmentation problem entails finding the boundary of an object in an image. The boundary of such an object may arise as a natural lower dimensional feature compared to the object and we combine topological and geometrical information in our method. To this end, we focus on the boundary persistence segmentation and active contour methods. The dimension reduction problem aims at finding a lower dimensional representation of the dataset by performing dimension reduction. We hope that, in the lower dimensional representation, the topological features remain. To address this problem, we focus on the circular coordinate method via persistence cohomology.