Olędzki, J.

Spiež, S.

Czech digital mathematics library

On embedding of curves into two-dimensional polyhedra

Institute of Mathematics AS CR, v. v. i.

WSAA 8J. Oledzki; Stanisƚaw SpieżOn embedding of curves into two-dimensional polyhedraIn: Zdeněk Frolík (ed.): Abstracta. 8th Winter School on Abstract Analysis.Czechoslovak Academy of Sciences, Praha, 1980. pp. 128--129.Persistent URL: http://dml.cz/dmlcz/701194Terms of use:© Institute of Mathematics of the Academy of Sciences of the Czech Republic,1980Institute of Mathematics of the Academy of Sciences of the Czech Republicprovides access to digitized documents strictly for personal use. Each copy of anypart of this document must contain these Terms of use.This paper has been digitized, optimized for electronic deliveryand stamped with digital signature within the project DML-CZ:The Czech Digital Mathematics Library http://project.dml.cz128 EIGHTH WINTER SCHOOL ON ABSTRACT ANALYSIS (1980) On embedding of curves into two-dimensional polyhedra 0, Oledzki and S. Spiez It is well known that curves (one-dimensional continua) can be embedded in the 3-dimensional Euclidean space. Some curves cannot be embedded into 2-dimensional polyhedra. For instance the Monger universal curve has such property, since each Its point has no neighborhood which is flat (can be em-bedded in a plane). Some curver which are not flat can be em-bedded into 2-dimensional (even 1-dimenslonal) polyhedra, A disc is a set homeomorphic to a closed ball in the plane. An n-book is the union of n discs such that the intersection of this discs is a segment lying in their boundary and any two of them have no other common point, R,M, Bing has noticed that a solenoid (the invers limit of circles) which is not flat can be embedded in a 3-book; a solenoid can be obtained here as the intersection of a decreasing sequence of the Mobius surfaces with added disc, everything lying in a 3-book, In shape theory it is known that for every curve X the-re exists a plane curve Y of the same shape as X if and only if X is movable. Our results are connected to the pro-blem (due to 0, Krasinkiewicz) what shapes are embeddable in an n-book, n.??3 • It is proved that every one-dimensional com-pact metric space with zero-dimensional set of non-locally flat points can be embedded into a 3-book, In particular for every curve there exists curve in a 3-book of the same shape. It is noticed that if X ; is one-dimensional compactum and for every point x€X there exists a neighborhood U , which can 129 be embedded in a 2-dimensional polyhedron then X can be embed-ded in a 2-dimensional polyhedron. 

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On embedding of curves into two-dimensional polyhedra

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Institute of Mathematics AS CR, v. v. i.