Connectedness Review

Algebraic Topology · Axiom Academy

Fundamental concepts of connectedness and path-connectedness in topological spaces 1. Connectedness and Disconnectedness A topological space X is connected if it cannot be written as a union of two non-empty disjoint open sets. Otherwise, X is disconnected . Connected: The real line ℝ, intervals [a,b], the circle S¹ Disconnected: The rationals ℚ, the union of two disjoint open intervals A space X is path-connected if any two points can be joined by a continuous path. This is a stronger condition than connectedness. Every topological space can be decomposed into maximal connected subspaces called connected components . Components partition the space X Each component is closed (but not necessarily open) Continuous maps send components to components If X is locally connected, components are also open

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