Connectedness

Simple connectedness

A path-connected space is simply connected¹ or 1-simply connected iff every continuous loop is null-homotopic, i.e. its fundamental group is trivial homotopy

𝜋1(𝑋,𝑥0)={𝑒}

for any 𝑥0 ∈𝑋.

Examples

• Every Star domain is simply connected.
• Fundamental group of a sphere

Properties

• Simply connectedness ⟹ Local simple connectedness ⟹ Semilocal simple connectedness

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tidy | en | SemBr

Footnotes

1. German einfach zusammenhängend ↩