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A metric condition which implies dimension $\leq 1$

Author(s): Michael Levin; Roman Pol
Journal: Proc. Amer. Math. Soc. 125 (1997), 269-273.
MSC (1991): Primary 54F45
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Abstract: A class of $1$ -dimensional spaces is distinguished by special type embeddings in compacta, or a corresponding metric property. In this setting, a simple proof of the Oversteegen-Tymchatyn theorem that the spaces of homeomorphisms of the Sierpi\'{n}ski's Carpet and the Menger Universal Curve have dimension $\leq 1$ is given.

References:

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3.: K.Kuratowski, Une application des images de functions á la construction de certains ensembles singuliers, Mathematica 6 (1932), 120-123.
4.: L.G.Oversteegen, E.D.Tymchatyn, On the dimension of certain totally disconnected spaces, Proc.A.M.S. 122(1994), 885-891. MR 95b:54040
5.: J.H.Roberts, The rational points in Hilbert space, Duke Math.Journ. 23 (1956), 489-492. MR 18:55b


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