engleski

Closed embeddings into Lipscomb's universal space

In this paper we prove that if X is a complete metrizable space of dim X < n+1 and weight not exceeding \tau, then there is a closed embedding of X into the subspace L_n(\tau) of Lipscomb's one-dimensional space J(\tau) consisting of points in J(\tau)^{;n+1}; with at least one irrational coordinate. Furthermore, any map from X to J(\tau)^{;n+1}; can be approximated arbitrarily close by a closed embedding of X into L_n(\tau). Also, relative and pointed versions are obtained. In the separable case an analogous result is obtained, in which the classic triangular Sierpinski curve (homeomorphic to J(3)) is used instead of J(\aleph_0).

Covering dimension; embedding; closed embedding; universal space; generalized Sierpinski curve; Lipscomb's universal space; extension; complete metric space

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