Continuous path

Given a Topological space $(𝑋, T)$ a continuous path or path in $𝑋$ is a continuous function $𝑐 : 𝕀 \to 𝑋$ , where $𝕀 = [0, 1]$ . Iff $𝑐$ is also a Embedding it is called an arc. topology A continuous path with the same start and endpoints is a Continuous loop.

Algebra

The set of paths $𝖳 𝗈 𝗉 (𝕀, 𝑋)$ may be made into a Magmoid $𝒫 𝑋$ with the concatenation operation. Let $𝛼 \in 𝒫 𝑋 (𝑥, 𝑦)$ and $𝛽 \in 𝒫 𝑋 (𝑦, 𝑧)$ . Then their concatenation $𝛽 ⊙ 𝛼 \in 𝒫 𝑋 (𝑥, 𝑧)$ is defined as

𝛽 ⊙ 𝛼 : 𝑡 \mapsto {𝛼 (2 𝑡) 𝑡 \in [0, 12] 𝛽 (2 𝑡 - 1) 𝑡 \in [12, 1]

Additionally, we have the involution of reverse path traversal: For $𝛼 \in 𝒫 𝑋 (𝑥, 𝑦)$ its reverse path $―― 𝛼 \in 𝒫 𝑋 (𝑦, 𝑥)$ is given by

―― 𝛼 : 𝑡 \mapsto 𝛼 (1 - 𝑡)

Clearly $𝒫$ defines a functor from Category of topological spaces to Category of magmoids Of more importance are the Category of paths and Fundamental groupoid, which are quotients modulo traversal and homotopy of paths respectively.

Properties

A path is just a homotopy of maps from the Single point space, i.e. of constant maps.

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Continuous path

Algebra

Properties

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