Continuity and Connectedness: A First Step
Luciano da Fontoura Costa · HAL (Le Centre pour la Communication Scientifique Directe) · 2021
Continuity and Connectedness constitute two central concepts in several theoretical and applied areas, including signal and image analysis, neuronal networks and deep learning, as well as scientific modeling in general. At the same time, several important theoretical concepts such as manifolds, which are extensively used in several applications, rely on continuity and connectedness. Because these two concepts can be studied from both the metric and topological perspective, it becomes particularly motivating to consider them from these two points of view. The present work aims at developing a brief introduction to continuity and connectedness from this integrated perspective. The metric approach is described first, followed by the topological perspective. Though most spaces typically found in applications are metric, the topological approach can provide interesting insights including path-connectedness and the definition of continuous functions without resourcing to distances. The particularly important topological concept of homeomorphism is also briefly discussed.