Paths and loops in a topological space

• This article aims towards defining the Fundamental group
• The name of the page was chosen to make it distinct from paths and loops, which are terms in graph theory

Path in a topological space

A path in $X$ is any continuous map $p:[0,1]\rightarrow X$.

Loop in a topological space

A path $p$ is a loop if $p(0)=p(1)$

Loop based at

If $p$ is a loop based at $x_0$ if $p(0)=p(1)=x_0$

Concatenating paths

Given two paths $p_0$ and $p_1$ in a topological space $X$ with $p_0(1)=p_1(1)$ we can obtain a new path by performing $p_0$ first, followed by $p_1$ in the same time by moving at double speed, this new path is called $p_0*p_1$ and is defined as:

• $$(p_0*p_1)(t)=\left\{\begin{array}{lr} p_0(2t) & \text{if }t\in[0,\tfrac{1}{2}] \\ p_1(2t-1) & \text{if }t\in[\tfrac{1}{2},1] \end{array}\right.$$

  this is fine to do as the functions agree when restricted to the intersection, that is to say when $t=\tfrac{1}{2}$ both "halves" agree

Note that:

• If $p_0$ and $p_1$ are loops based at $x_0$ then $p_0*p_1$ is always defined and is itself a loop based at $x_0$
• $p_0*p_1\ne p_1*p_0$ usually - which is really easy to see.
• We can't define a group yet.
  • Note that $a*(b*c)$ performs $b$ and $c$ at 4 times their normal speed and $a$ at just double whereas:
  • $(a*b)*c$ performs $a$ and $b$ at 4x their normal speed and $c$ at just double - these are clearly different paths - so we don't even have associativity

Constant path

The constant path (often denoted $e:[0,1]\rightarrow X$ is a map where:

• $$\forall t\in[0,1]$$ we have $$e(t)=x_0$$

Clearly this is a loop