Confusion about the definition of a "loop" in a topological space

Question

A continuous map $f:[0,1] \to X$ is called a path and if $f(0)=f(1)$ then it is called a loop.

But any loop looks like a circle which is not a function as it is not well defined. How did it possible? Can someone explain me please?

Thanks in advance.

Answer 1

You are confusing things.

You are correct that there is no function $f: \mathbb{R} \to \mathbb{R} $ such that the points $(x,f(x))$ (over any set of $x$) describe a circle.

However, you can define a function $g: \mathbb{R} \to \mathbb{R}^2$ whose locus defines a circle, to wit, $g(t) = (\cos t, \sin t)$.

Answer 2

A loop is terminology for a special class of functions $f:[0,1]\to X$, namely those continuous functions such that $f(0)=f(1)$. Generally, if we have a continuous function $f:[0,1]\to X$, then $f$ is called a path. A generalization of the idea of a parametric function, we call such a function a path because as $t$ goes from $0$ to $1$, $f(t)$ traces the "curve" starting at $f(0)$, following along $f([0,1])$, and ending at $f(1)$. In the special case that $f(0)=f(1)$, the path ends where it began, which is what you'd expect from something called a "loop."