Path connectedness seems to be defined in a topological space, but can the existence of a path be proven without using the functions of vector addition, scalar multiplication and norm ?
For example, $R^n$ is a metric space (with the Euclidean metric) and has a corresponding induced topology. But to prove that a ball (open or closed) is path connected I would construct a mapping from $ t \in [0, 1]$ to $R^n$ defining a path between $a, b \in R^n$ by $y = a + t (b - a)$ and then use the norm to show that this is continuous and within the ball. In this case (and in general) can one prove path connectedness using only metric or topological concepts ?
After comments and a couple of answers, perhaps I understand my question better. It was prompted particularly in consideration of $R^n$. It seems to be confirmed (by comments) that one cannot prove the existence of a continuous path in $R^n$ without defining a function which relies on vector addition, scalar multiplication and norm.
Path connectedness is defined for a topological space and $R^n$ induces a topology and has additional metric properties, but requires yet more properties (normed vector space) in order to prove path connectedness. This strikes me as odd because normally when a concept is defined in a mathematical structure one would expect to be able to prove various results within that structure. In this case I would have expected some, perhaps non-constructive proof, of existence of a path: apparently not.