Appropriateness of using an interval in $\Bbb{R}$ as the parameter to a continuous path in space $X$. The fundamental group $\pi_1(X,x)$ is usually defined using continuous paths $p : [0,1] \to X$.  But...
(1) Can you use other spaces besides a closed interval of $\Bbb{R}$ in the usual topo?
(2) How will the resulting fundamental group change if at all.
(3) Do there exist spaces $X$ where a closed interval of $\Bbb{R}$ in the usual topo doesn't really work for some reason, like $X$ has "too many points"?
Thanks.
 A: You can indeed use other objects instead of a closed interval. The important thing to make things work is when considering a homotopy between two morphisms $f,g:X\to Y$ is to replace $Y$ by a cylinder object for it. A cylinder object for $Y$ is an a space (and more generally the situation can be considered in any Quillen model category, so in fact the situation is much much broader), typically denoted by $Y\wedge I$ (though it does not have to be related to any kind of binary operation or an interval, it's just another object) such that, intuitively, it contains two copies of the original $Y$ and it contracts to any of these two copies. (for forma details see any text on Quillen model categories (e.g., Hovey, Hirschhorn, Jardin, etc (I particularly like Hirschhorn's). When thinking about it, this is why a closed interval works, it contracts nicely to a point so when considering $Y\times [0,1]$ you get two copies of $Y$ in it and everything collapses continuously back to any of these copies. The theory of homotopy groups in Quillen model categories resembles the usual one quite closely. 
There are however spaces for which this formalism won't work. For instance, analogously to the long line you can consider a long circle $C$ (e.g., take a two-point compactification and then identify the two points at infinity). For such a space any path $[0,1]\to C$ is contractible. In fact any continuous function $\mathbb R\to C$ is contractible. After all, it's the long circle, so $\mathbb R$ is simply too short. However, intuitively, the long circle should also have fundamental group isomorphic to $\mathbb Z$. This can be done by considering long (enough) paths. 
There are other spaces that exhibit problematic behaviour with respect to paths parametrized by closed intervals. For instance, any such path in $\mathbb Q$ is constant, and thus the rational circle has trivial fundamental group. That too can be remedied by considering paths parametrized by rational intervals. It seems though that the most important cases are for spaces that are nice enough that no such pathologies arise in them. After, the homotopy groups of spheres are still a big mystery. Before we delve into obscure spaces, we'd better understand spheres. 
