Abstract
By a sin (1/x)-curve is meant a metric continuum that is a 1-1 continuous image of the disjoint union of an arc and a semi-open interval that has the image of the arc as continuum of convergence. It is shown that ifM is a compact metric space,A ⊂M an arc, whileM/A is an arc havingA/A as an end-point, thenM is an arc, a triod, some sin (1/x)-curve, or some sin (1/x)-curve with an arc attached at one point, or some sin (1/x)-curve with two arcs attached. The case of shrinking finitely many arcs is also considered in an attaching theorem.
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Prepared under a NASA Research Grant No. NsG-568 at Kent State University.
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Doyle, P.H. On shrinking arcs in metric spaces. Israel J. Math. 5, 104–106 (1967). https://doi.org/10.1007/BF02771629
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DOI: https://doi.org/10.1007/BF02771629