Lavrentieff's continuation

The continuation of Lavrentieff is a theorem from the mathematical branch of topology , which goes back to the Russian mathematician Michail Lavrentieff . It is related to Mazurkiewicz's theorem and deals with a continuation property of complete metrical spaces .

Formulation of the sentence

Given are complete metric spaces and and therein subspaces and and a homeomorphism . Then: ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle A\subseteq X}$ ${\displaystyle B\subseteq Y}$ ${\displaystyle h\colon A\rightarrow B}$

There are lots ${\displaystyle G_{\delta }}$

${\displaystyle A^{*}\subseteq X}$ and ${\displaystyle B^{*}\subseteq Y}$

With

${\displaystyle A\subseteq A^{*}\subseteq {\overline {A}}}$ and ${\displaystyle B\subseteq B^{*}\subseteq {\overline {B}}}$

plus a homeomorphism

${\displaystyle h^{*}\colon A^{*}\rightarrow B^{*}}$ ,

which is a continuous continuation of . ${\displaystyle h\colon A\rightarrow B}$

swell

• M. Lavrentieff: Contribution à la théorie des ensembles homéomorphes . In: Fundamenta Mathematicae . tape 6 , 1924, pp. 149-160 .
• Jürgen Heine : Topology and Functional Analysis . Basics of abstract analysis with applications. R. Oldenbourg Verlag , Munich 2011, p. 218 .
• Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. 1970, p. 178 .

Web link

Lavrentieff's continuation in the Encyclopedia of Mathematics (online)