CONSTRUCTING ARCS FROM PATHS BY COLLAPSING SUBLOOPS

It is a well-known fact that every path-connected Hausdorff space is arcwise connected. Typically, this result is viewed as a consequence of a sequence of fairly technical results from continuum theory. In this expository note, we present a direct proof of this statement, which constructs an arc from a path by “deleting” a maximal collection of subloops. By breaking the proof into multiple steps, we examine generalizations of this result and the limits of our line of argument. In particular, we identify a modest extension to a class of spaces relevant to algebraic topology.