Definition

Connectedness and Path-Connectedness

Lie Groups and Lie Algebras · lie-groups.tex

A topological space $X$ (Definition~Topological Space) is connected if it cannot be written as a disjoint union of two nonempty open sets. It is path-connected if for every $x, y \in X$ there exists a continuous path $\gamma : [0,1] \to X$ (Definition~Continuous Map and Homeomorphism) with $\gamma(0) = x$ and $\gamma(1) = y$.

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Dependency Graph

flowchart LR classDef current fill:#6366f1,color:#fff,stroke:#4f46e5 nc9093e38["Topological Space"] na9fa06fb["Continuous Map and Homeomorphism"] ndc4990c2["Connectedness and Path-Connectedness"]:::current n5592bb9e["Simply Connected"] n738e8542["Universal Cover of a Lie Group"] nc9093e38 --> ndc4990c2 na9fa06fb --> ndc4990c2 ndc4990c2 --> n5592bb9e ndc4990c2 --> n738e8542 click nc9093e38 "../objects/c9093e38.html" "_self" click na9fa06fb "../objects/a9fa06fb.html" "_self" click n5592bb9e "../objects/5592bb9e.html" "_self" click n738e8542 "../objects/738e8542.html" "_self"