Math 2.0

Algebraic topology Definition

Simply connected domain

Builds on Closed curve

Definition

Let X⊆ℝnX \subseteq \mathbb{R}^n. The set XX is simply connected if XX is path-connected and each continuous closed curve c ⁣:[0,1]→Xc \colon [0, 1] \to X has this property. There are a point p∈Xp \in X and a continuous map H ⁣:[0,1]×[0,1]→XH \colon [0, 1] \times [0, 1] \to X such that H(t,0)=c(t),H(t,1)=p,H(0,s)=H(1,s)for all s,t∈[0,1].H(t, 0) = c(t), \qquad H(t, 1) = p, \qquad H(0, s) = H(1, s) \quad \text{for all } s, t \in [0, 1]. An open simply connected set is a simply connected domain.

Remarks

The map HH deforms the loop cc into the constant loop at pp through closed loops t↦H(t,s)t \mapsto H(t, s) in XX. Each convex set is simply connected. For example, ℝn\mathbb{R}^n and each ball are simply connected. The plane minus the origin is not simply connected, and neither is ℝ3\mathbb{R}^3 minus the zz-axis. But ℝn\mathbb{R}^n minus a point is simply connected for n≥3n \ge 3. An equivalent condition: XX is path-connected, and any two paths in XX with the same end points can be deformed into each other with the end points fixed.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionClosed curve, definition, no evidence levelClosed curvedefinitionSimply connected domain, definition, no evidence levelSimply connected doma…definition
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  11. Higher partial derivatives definition
  12. Iterated derivative definition
  13. C^k function characterised by continuous derivatives statement
  14. Parametrised curve definition
  15. Closed curve definition
  16. Simply connected domain definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Simply_connected_space, Definition and equivalent formulations
  2. Web https://en.wikipedia.org/wiki/Simply_connected_space, Examples
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