Geodesic
A Note On Arc-Preserving Functions For Manifolds1
Canadian mathematical bulletin, Tome 10 (1967) no. 4, pp. 597-598

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Hall and Puckett [2] have shown that an arc - preserving function defined ona locally connected continuum having no local separating points is ahomeomorphism if its total image is not an arc or point. This note showsthat their results can be extended to non-compact manifolds. 
Charlton, H. J. A Note On Arc-Preserving Functions For Manifolds1. Canadian mathematical bulletin, Tome 10 (1967) no. 4, pp. 597-598. doi: 10.4153/CMB-1967-059-2
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[1] 1. Doyle, P. H. and J. G. Hocking, , A decomposition the or em for n-dimensional manifolds. Proc. Amer. Math. Soc. 13 (1966), 469-471. Google Scholar

[2] 2. Hall, D. W. and Puckett, W. T. Jr., Conditions for continuity of arc preserving transformations. Bull. Amer. Math. Soc. 47 (1944), 468-475. Google Scholar

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