Is the wedge sum of circles considered as a path?
According to Wikipedia(Rose), the figure eight graph $S^1\vee S^1$ is directed even though it is the direct sum of two circles with two points equivalent
2026-03-25 14:32:28.1774449148
Wedge sum of circles
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The second picture there has directed loops just for illustrating that they are the generators of the fundamental group of figure eight. Without such specific purpose, you don't need to understand any topological space to be directed.