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showed in 1878. Cantor's result came as a surprise to many mathematicians and kicked off the line of research leading to space-filling curves, Osgood curves, and Netto's theorem. A near-bijection from the unit square to the unit interval can be obtained by interleaving the digits of the decimal
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and a two-dimensional set, see
Section 6.4, "Proof of Netto's Theorem", pp. 97–98. For the application of Netto's theorem to self-intersections of space-filling curves, and for Osgood curves, see Chapter 8, "Jordan Curves of Positive Lebesgue Measure", pp.
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from one-dimensional spaces to two-dimensional spaces. They cover every point of the plane, or of a unit square, by the image of a line or unit interval. Examples include the
109:. The conditions of the theorem can be relaxed in different ways to obtain interesting classes of functions from one-dimensional spaces to two-dimensional spaces:
128:. Neither of these examples has any self-crossings, but by Netto's theorem there are many points of the square that are covered multiple times by these curves.
234:. For the statement of the theorem, and historical background, see Theorem 1.3, p. 6. For its proof for the case of bijections between the
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An important special case of this theorem concerns the non-existence of continuous bijections from one-dimensional spaces, such as the
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Dauben, Joseph W. (1975), "The invariance of dimension: problems in the early development of set theory and topology",
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of points in the square. The ambiguities of decimal, exemplified by the two decimal representations of 1 =
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in the plane. However, by Netto's theorem, they cannot cover the entire plane, unit square, or any other
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If one relaxes the requirement of continuity, then all smooth manifolds of bounded dimension have equal
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in 1879, gave faulty proofs of the general theorem. The faults were later recognized and corrected.
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The case for maps from a higher-dimensional manifold to a one-dimensional manifold was proven by
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are continuous bijections from one-dimensional spaces to subsets of the plane that have nonzero
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rather than a bijection, but this issue can be repaired by using the
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82:can be mapped continuously and bijectively to the
78:to show that no manifold containing a topological
20:The first three steps of construction of the
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207:, Universitext, New York: Springer-Verlag,
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101:, to two-dimensional spaces, such as the
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300:The American Mathematical Monthly
118:surjective continuous functions
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293:GouvĂŞa, Fernando Q. (2011),
269:10.1016/0315-0860(75)90066-X
155:cardinality of the continuum
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176:Schröder–Bernstein theorem
86:. Both Netto in 1878, and
76:intermediate value theorem
213:10.1007/978-1-4612-0871-6
295:"Was Cantor surprised?"
162:representations of the
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144:two-dimensional region
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164:Cartesian coordinates
42:mathematical analysis
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353:Theorems in topology
256:Historia Mathematica
204:Space-filling curves
201:Sagan, Hans (1994),
114:Space-filling curves
74:in 1878, using the
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159:Georg Cantor
138:. They form
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88:Georg Cantor
72:Jacob LĂĽroth
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48:states that
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34:Osgood curve
262:: 273–288,
151:cardinality
122:Peano curve
107:unit square
65:Eugen Netto
342:Categories
183:References
53:bijections
50:continuous
172:injection
95:real line
84:real line
61:dimension
59:preserve
239:131–143.
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278:0476319
231:1299533
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