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. #m/def/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 $𝖳 𝗈 𝗉$ to $𝖬 𝖺 𝗀 𝖽$ . 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 continuous maps from the Single point space, i.e. of constant maps.

#state/tidy | #lang/en | #SemBr