Base B for X: Connected Sets and Open Bases

This lecture discusses the concept of a base B for a topological space X, focusing on open sets that are connected and form a base for X. It explores the properties of X being connected by arcs and locally connected by arcs, emphasizing the importance of having an open base. The lecture also covers the existence of paths between points in X and the significance of not having a base point. Additionally, it delves into the proof related to the triviality of certain components in X and the implications of different topological properties.