I've been reading An Introduction to Algebraic Topology by Andrew Wallace.
His definition of an arcwise connected space is: a space in which there is a path between every two points.
In the Wikipedia page for connected space, however, this is taken as the definition of a path-connected space not an arcwise connected space.
I am confused by this mix-up. Which one is more standard?
I'm quoting Wallace's definition of path and arcwise connectedness.
Definition. Let $E$ be a given topological space, and let $I$ denote the unit interval $0 \le t \le 1$, regarded as a subspace of the space of real numbers in the usual topology. Then a path in $E$ joining two points $p$ and $q$ of $E$ is defined to be a continuous mapping $f$ of $I$ into $E$ such that $f(0) = p$ and $f(1) = q$. The path will be said to lie in a subset $A$ of $E$ if $f(I) \subset A$.
Definition. A topological space $E$ is said to be arcwise connected if, for every pair of points $p$ and $q$ of $E$ there is a path in $E$ joining $p$ and $q$. If $A$ is a set in a topological space $E$, then $A$ is arcwise connected if every pair of points of $A$ can be joined by a path in $A$.
It seems that the situation is as Henno Brandsma pointed out: since most spaces are Hausdorff, path-connectedness is identified with arcwise connectedness and thus entirely omitted from the book. However, this strikes me as odd as I am actually interested in non-Hausdorff spaces such as those arising from computer science.