their usual topology. The Menger–Nöbeling theorem (1932) states that if is compact metric separable and of dimension , then it embeds as a subspace of Euclidean space of dimension . (Georg Nöbeling was a student of Karl Menger. He introduced Nöbeling space, the subspace of consisting of points with at least co-ordinates being irrational numbers, which has universal properties for embedding spaces of dimension .) Assuming only X metrizable we have (Miroslav Katětov) ind X ≤ Ind X = dim X; or assuming X compact and Hausdorff (P. S. Aleksandrov) dim X ≤ ind X ≤ Ind X. Either inequality here may be strict; an example of Vladimir V. Filippov shows that the two inductive dimensions may differ. A separable metric space X satisfies the inequality if and only if for every closed sub-space of the space and each continuous mapping there exists a continuous extension . References Further reading Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 844-55. R. Engelking, Theory of Dimensions. Finite and Infinite, Heldermann Verlag (1995), . V. V. Fedorchuk, The Fundamentals of Dimension Theory, appearing in Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I, (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin . V. V. Filippov, On the inductive dimension of the product of bicompacta, Soviet. Math. Dokl., 13 (1972), N° 1, 250-254. A. R. Pears, Dimension theory of general spaces, Cambridge University Press (1975). Dimension theory