Please note that I am a beginner in Algebraic Topology. I am struggling with the intuition of homotopy equivalence in non-Euclidean spaces. To be more specific, we know that any two loops with the same base point in $\mathbb{R}^n$ are equivalent. This is because we can, intuitively, shrink down the first loop to the base point, and then expand it to the second loop.
Here is my proof: Let $\alpha$ and $\beta$ be two loops in $\mathbb{R}^n$ that share a base point $p$. We can construct a homotopy between $\alpha$ and $p$ via $$H(x,t)=t\alpha(x) + (1-t)p$$ We can do the same with $\beta$ and then we see that $\alpha \simeq p \simeq \beta$, so $\alpha$ and $\beta$ are homotopic.
I struggle to grasp the idea that two loops can share a basepoint and not belong in the same homotopy class. I think this is because we "visualize" in $\mathbb{R}^n$ (with $n \le 3$). Can someone provide an example of two loops that are not homotopic but share a base point? Perhaps in a path-connected space? Futhermore, can someone provide tips and perhaps a space to visualize when working with abstract homotopy classes? Thank you.
So, here is an easy way of looking at it: imagine you've got some infinitely strechy string. Now, go outside to your local neighbourhood (literal, not topological), nail down an end of the string to the street, and take the other end with you. Now go on a long and twisty journey across the neighbourhood - no backtracking, just a long walk where you go around many buildings and come back to where you started at the end. Now, nail the other end of the string to the same spot as the first spot is nailed to.
So, the giant string is now essentially just physically tracks the trail that you walked. Keeping in mind all the buildings in between, can you pull the string back to a small clump under you without breaking it or detaching the nails?
Unless you lift them over the buildings, the answer is probably no.
So, to return to the abstract land of topology: take $\mathbb{R}^2$. To make a visual model of the scenario we just outlined, let $(0,0)$ be the basepoint, and remove from the set whichever points correspond to buildings in your neighbourhood. It probably isn't hard to see that loops with basepoint $(0,0)$ are mathematical versions of the physical example I just outlined. If a loop goes around a building - which, it would be more apt to call a hole in our case - then it cannot be reduced to a point.
Edit: since you asked for an example, by far the easiest example to visualize is a cylinder. If a loop goes around the cylinder, it cannot be contracted to a point.