Relative and pointed versions of Lipscomb's embedding theorem
by
Uroš Milutinović
University of Maribor
Coauthors: Ivan Ivanšić (University of Zagreb)

In his papers [3, 4] S. L. Lipscomb defined the space J(\tau) as a factor-space of Baire's universal 0-dimensional space and proved that any n-dimensional metrizable space of weight \tau, \tau >= \aleph₀, can be embedded in the subspace L_n(\tau)={x in J(\tau)ⁿ⁺¹ :at least one coordinate of x is irrational} of J(\tau)ⁿ⁺¹.

In an attempt to prove the relative version of the theorem, i.e. to prove that any embedding f₀:X₀ --> L_n(\tau), where X₀ is a subspace of a n-dimensional metric space X of weight \tau, can be extended to an embedding f:X --> L_n(\tau), we found simple examples showing that this is in general not true.

We succeeded to prove such a theorem for n=0 and compact X₀ [2]. But for n > 0 even the case when X₀ is a single point is not trivial.

Using the fact that J(\tau) is naturally homeomorphic to a generalized Sierpi\' nski curve [5, 6] and techniques of modification of Lipscomb's decompositions and indexing of the modified decompositions developed in [1, 6], here we prove the pointed version of Lipscomb's embedding theorem, i.e. we show that the embedding may be chosen in such a way that its value is given in advance at a certain point (the base point). This is not trivial precisely because J(\tau) splits into the rational and the irrational part.

References

[1] I. Ivansi\'c and U. Milutinovi\' c. A universal separable metric space based on the triangular Sierpi\'nski curve. Top. Appl. 120 (2002) 237-271.

[2] I. Ivansi\'c and U. Milutinovi\' c. Relative embeddability into Lipscomb's 0-dimensional universal space. Houston J. Math. (to appear)

[3] S. L. Lipscomb. A universal one-dimensional metric space. In TOPO 72 - General Topology and its Applications, Second Pittsburgh Internat. Conf., volume 378 of Lecture Notes in Math. Springer-Verlag, New York, 1974, 248-257.

[4] S. L. Lipscomb. On imbedding finite-dimensional metric spaces. Trans. Amer. Math. Soc., 211 (1975) 143-160.

[5] S. L. Lipscomb and J. C. Perry. Lipscomb's L(A) space fractalized in Hilbert's l²(A) space. Proc. Amer. Math. Soc., 115 (1992) 1157-1165.

[6] U. Milutinovi\' c. Completeness of the Lipscomb universal space. Glas. Mat. Ser. III, 27(47) (1992) 343-364.

Date received: August 30, 2002