Abstract
Let n = 1, 2, 3, ... Denote by C n the set of those “points”, for which 0 ≦ x i ≦ 1 (1 ≦ i ≦ n). The set C n is the simplest example of a point set of dimension n. In particular, for m < n, the dimension of the set C m is less than the dimension of the set C n . The question on the precise mathematical meaning of the italicized statement is, however, not quite easy. On the first sight it may seem that C n has more points than C m . But as soon as in 1877, G. Cantor showed1) that this is not the case, that there exists a one-to-one mapping of C m onto C n Try then otherwise! It is clear that (still for m < n) C m is a continuous image of the set C n in comparison with the opinion that C n is not a continuous image of the set C m . But even this way does not lead to the aim, because in 1890, G. Peano2) showed in turn that there exists a continous mapping of the set C m onto the set Cn, too.
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References
Journal f. Math., vol. 84 (1877), p. 242.
Math. Ann., vol. 36 (1890), p. 257.
Math. Ann., vol. 70 (1911), p. 166. The proof indicated here is not complete and Lebesgue gave a precise exposition only in Fund. Math., vol. 2 (1921), p. 256. The first precise proof of Lebesgue’s lemma was given by Brouwer in the paper quoted in the remark 5). Lebesgue’s proof was later substantially simplified by W. Hurewicz [Math. Ann., vol. 101 (1929), p. 210] and by E. Sperner [Hamb. Abh., vol. 6 (1928), p. 265].
The first precise proof of this fact was given by Brouwer in Math. Ann., vol. 70 (1911), p. 161.
Journ. f. Math., vol. 142 (1913), p. 210 [see also vol. 153 (1924), p. 253].
A systematic exposition of the theory is given by K. Menger in his book Dimensionstheorie (1928). See also posthumously published Urysohn’s treatise in Fund. Math., vol. 7 (1925), pp. 30–137 and vol. 8 (1926), pp. 225–351. The paper by V. Jarník in Časopis, vol. 58 (1929), p. 367, can help well for the first information.
E.g., at problems studied in Chap. 4. in Menger’s book.
Rozpr. II. tř. čes. Akad., vol. 42, 13 (1932). See also Comptes Rendus Paris, vol. 193 (1931), pp. 976–977.
See head 25.
See head 26.
Fund. Math., vol. 8, p 301.
See heads 23 and 24.
See head 9.
The beginner reader will perhaps lack the proof of this fact. I shall have an opportunity to present such a proof in another paper, which will soon appear in this Journal.
Annals of Math., vol 30 (1929), p. 120 (Überführungsatz). An easy proof will appear in the paper by C. Kuratowski Sur un théorème fondamental etc. in vol. 20, Fund. Math.
Časopis, vol. 61 (1932), p. 109. Quoted under an abbr. M.
M, I.
M, 2).
M, 3.
M, 6).
S is a topological space by M, 5.
M, 5.
M, 8).
29)M,7).
The properties 5.4 and 5.5 are usually included into the definition of metric space. Deducing them from the properties is due to A. Lindenbaum in Fund. Math., vol 8 (1926), p. 211.
Neither the set A nor r need to be uniquely determined by the set K(A,r). An example: Let the space R contain only two points a, b and let g(a, b) = 1. Then K(a, 2) = K(b, 3).
This notion appears for the first time in H. Tietze [Math. Ann., vol. 88 (1923), p. 301];
the name normal in P. Urysohn [Math. Ann., vol. 94 (1925), p. 265].
M, 4.
W. Hurewicz and K. Menger, Math. Ann., vol. 100 (1928), remark 22).
The number of all combinations of indices (Math) After all, for our purposes it is essential only that this number is finite. 36) M, 1,3.
K. Menger, Dimensionstheorie, p. 159–160 (Bemerkung).
Following the manuscript of Topologie by Kuratowski. (It will appear in the series Monografie Matemaiyczne in vol. 3.)
M, 4)
Theorems 18, 19 and 27 are due to Urysohn; quoted in 32), pp. 285–288.
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© 1993 Miroslav Katětov, Petr Simon et al.
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Čech, E. (1993). Contribution to Dimension Theory. In: Katětov, M., Simon, P. (eds) The Mathematical Legacy of Eduard Čech. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7524-0_12
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