I am having difficulty in distinguishing between two equivalence classes of the fundamental group at a base pont $x_0$ of a topological space $X$. Given $X$ and an arbitrary point $x_0\in X$ one defines homotopy as equivalence relation on the set of functions on $X$ with base point $x_0$ called loops. In other words, all loops at a fixed base point $x_0$ seem to me to be homotopic, and thus belonging to one equivalence class, say [f], since two such loops seem to be continuously deformable into one another.
Can somebody help me understand to discriminate between two equivalence classes, [f] and [g], modulo the homotopy of loops at a base point $x_0$ ?
Many thanks.
Imagine the punctured plane $\Bbb R^2\setminus \{0\}$. Any loop that encircles the origin is not homotopic to a loop that doesn't contain the origin, though they may share the same base point. Intuitively, any loop that doesn't contain the origin can be shrunk to a point because there is no hole to obstruct it during its homotopy, but loops that do encircle the origin can't be shrunk to a point because we would have to "tear" them at some point in the homotopy.
This isn't precise, but you're studying algebraic topology to learn how to make it precise. More precisely, we would say that $\pi_1(\Bbb R^2\setminus\{0\},(1,0))\cong \Bbb Z$ is a nontrivial group that is generated by an element $[f]$ (such as a loop that encircles the missing origin). Any loop $g$ that does not encircle the origin is nulhomotopic, so its homotopy class $[g]$ is trivial, so we can't have $[f]=[g]$ because $\Bbb Z$ is a nontrivial group.