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- W2330019409 abstract "In 1921, S. Mazurkiewicz [1] raised the question: if a bounded continuum in a space of m dimensions is topologically equivalent to each of its nondegenerate subcontinua, is it an arc? In 1948, making use of a certain plane indecomposable continuum, E. E. Moise [2] showed that this question may be answered in the negative. In this paper it is shown that for compact decomposable continua in a metric space, the answer is affirmative. At the outset the space S is assumed to be compact and metric. After Theorem 5 it will be convenient to place further restrictions on S. An arc is the homeomorphic image of the number interval [0, 1]. A continuum is said to be decomposable if it is the sum of two continua neither of which is M. An arc is, of course, decomposable. If a continuum is not decomposable, it is said to be indecomposable. A decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua will be called an A-continuum. (The terminology in this paper is similar to that used in R. L. Moore's Foundations of Point Set Theory [3]). If x and y are distinct points of a continuum M, some subcontinuum of M is irreducible from x to y [3, Theorem 34, Ch. I]. Hence if M is an A-continuum, M itself must be irreducible between some two of its points." @default.
- W2330019409 created "2016-06-24" @default.
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- W2330019409 date "1960-11-01" @default.
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- W2330019409 title "Proof That Every Compact Decomposable Continuum Which is Topologically Equivalent to Each of Its Nondegenerate Subcontinua is an Arc" @default.
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- W2330019409 doi "https://doi.org/10.2307/1970224" @default.
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