Path - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A path in a topological space $X$ is a continuous map $\gamma\colon [0,1]\to X$ , where $[0,1]$ is the interval with its usual topology. The points $\gamma(0)$ and $\gamma(1)$ are called the initial and terminal points of the path.

Examples

In $\mathbb{R}^2$ , for points $a,b\in\mathbb{R}^2$ , the map $\gamma(t)=(1-t)a+tb$ is a path from $a$ to $b$ .
The map $\gamma(t)=(\cos(2\pi t),\sin(2\pi t))$ is a path in the unit circle with $\gamma(0)=\gamma(1)$ (a loop).