Vignettes. Part 3.

A usual convention in topology is to have arc mean path in a space which also is a homeomorphism onto its image. This is simultaneously a felicity and a hazardousness =infelicity.

It is a hazardousness in that in combinatorics it is absolutely standard to have path signal that no repetitions whatsoever are allowed, while in topology path tends to signal that self-intersections are allowed. It is a felicity in that it is absolutely standard to call the edges of (abstract) digraphs by the name arc, and it is usual to formalize embedded directed graphs (embedded in some space or the other, e.g. in \mathbb{R}^2 to get plane digraph, or in \mathbb{H}^2 to get hyperbolic digraphs) by arcs in the usual topological, self-intersections-forbidden sense of the word. So in that sense, arc and arc correspond felicitously.

( And by the way, although we defined the term hazardousness classically, by letting hazardousness := infelicity, we have just seen, by example, that it is not true that  [hazardousness    \Rightarrow      \neg felicity ]. Fully spelled out

\neg [infelicity    \Rightarrow      \neg felicity ]

To sum up, there are terms which are simultaneously a hazardousness and a felicity. Phew.

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