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Clarified the definition used for an arc.
Ian Iscoe
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The desired result appears in M.H.A. Newman's classic book, Elements of the Topology of Plane Sets of Points (2nd ed., 1951), as Theorem 14.5 in Chapter VI, on pg. 164:

Theorem 14.5: Every simple arc in $X^2$ is an arc of a simple closed curve in $X^2$.

$X^2$ is Newman's notation for a space that is either the "open" plane, $R^2$, or the "closed" plane, $R^2 \cup \{\infty\}$.

N.B. The proof of Theorem 14.5 does not involve Schoenflies' Theorem. Also, Newman's definition of an arc is such that it has ends.

Ian Iscoe
  • 311
  • 2
  • 5