Weak chainability of pseudocones

Bellamy, David P.

doi:10.1090/S0002-9939-1975-0365515-9

Weak chainability of pseudocones
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by David P. Bellamy

Proc. Amer. Math. Soc. 48 (1975), 476-478

DOI: https://doi.org/10.1090/S0002-9939-1975-0365515-9

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Abstract:

A pseudocone over $X$ is a compactification of $(0, 1]$ with remainder $X$. $S$ is a circle. A characterization of those pseudocones over $S$ which are weakly chainable is given. (A continuum is weakly chainable if and only if it is a continuous image of the pseudoarc.) Covering projections and liftings are used, and a simple geometric interpretation of the result is that a pseudocone over $S$ is weakly chainable if and only if the absolute value of the winding number of any subarc of $(0, 1]$ around $S$ is bounded by some $m > 0$.

References

Topological properties of compactifications of a half open interval

David P. Bellamy, An uncountable collection of chainable continua, Trans. Amer. Math. Soc. 160 (1971), 297–304. MR 286076, DOI 10.1090/S0002-9947-1971-0286076-X
M. K. Fort Jr., Images of plane continua, Amer. J. Math. 81 (1959), 541–546. MR 106441, DOI 10.2307/2372912
A. Lelek, On weakly chainable continua, Fund. Math. 51 (1962/63), 271–282. MR 143182, DOI 10.4064/fm-51-3-271-282
J. Mioduszewski, A functional conception of snake-like continua, Fund. Math. 51 (1962/63), 179–189. MR 144313, DOI 10.4064/fm-51-2-179-189
Sam B. Nadler Jr. and J. Quinn, Embeddability and structure properties of real curves, Memoirs of the American Mathematical Society, No. 125, American Mathematical Society, Providence, R.I., 1972. MR 0353278
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112