Extension of non-expanding functions and metrics on metric and ultrametric spaces
Alex J. Lemin
Moscow State University of Civil Engineering

We discuss the Extension Problem for non-expanding functions and metrics defined on metric and ultrametric spaces. Let METR(c) denote a category of all metric spaces of diameter $\leq c$ and non-expanding maps, and METR$^{\circ }$ denote a category of all metric spaces with a base point and non-expanding bounded maps that take a base point to a base point. For any c>0, denote by I a closed segment [0,c]; consider a real line $% {\bf R}$ with a base point 0. The sets N(X,I) and $N(X,{\bf R})$ of all morphisms from X to I (to ${\bf R}$, respectively) are endowed with a metric of uniform convergence.

Theorem 1. For any subset A in X and any f in N(A,I) (in $N(X,{\bf R})$, respectively) there exists a map F in N(X,I) (in $N(X,{\bf R})$, respectively) such that F|_A=f. Moreover, there exists a greatest element $F_{\max }$ and a smallest element F$_{\min }$ in a set of such extensions. Theorem 1 is an analogous of the Titze-Uryson Extension Theorem.

Theorem 2. Both the sets N(X,I) and $N(X,{\bf R})$ are lattices. The lattice N(X,I) is complete, Brouwerian and dually Brouwerian. The lattice $N(X,{\bf R})$ is conditionally complete, Brouwerian and dually Brouwerian.

Theorem 3. Any uniformly continuous bounded function can be uniformly approximated by linear combinations of non-expanding functions. This is an analogous for the Weierstrass-Stone Approximation Theorem. By the Kuratowski Theorem any X in METR(c) can be embedded isometrically in N(X,I). Is it true that any X in METR$^{\circ }$ can be embedded isometrically in $N(X,{\bf R})$?

Theorem 4. The space N(X,I) is isometric to a metric power I$^\tau$ (a product of $\tau$ copies of the segment I in METR(c)). The space $N(X,{\bf R})$ is isometric to a pointed metric power ${\bf R}$$^\tau$ (a product of $\tau$ copies of ${\bf R}$ in METR$^{\circ }$). Here $\tau$ denotes a weight of X.

Theorem 5. For any subset A in a metric space (X,D) and any metric d on A no greater than D|_A, there is a metric $d^{\prime }$ on X that extends d. Moreover, there exists a greatest of such extensions. An assignment $X\rightarrow N(X,I)$ ( $X\rightarrow N(X,{\bf R})$, respectively) is a contra-variant functor fromMETR(c) to METR(c) (from METR$^{\circ }$ to METR$^{\circ }$, respectively). We describe its properties, in particular, its action on various types of morphisms (momomorphisms, epimorphisms, regular monomorphisms, equalizers, etc). For ultrametric spaces, more or less close analogous' of the Theorems above can be proved. E.g., any non-expanding map f from a subset A in an ultrametric space X to a two-point space $% \{0,c\}$ can be extended to X.

Theorem 6. For any ultrametric space (X,d) there exists a family $\{$ $T_\alpha \vert\alpha \in \Gamma \}$ of two-point spaces $% T_\alpha =\{0,c_\alpha \}$ such that (X,d) is isometric to a subspace of their metric product and a number of factors in the product is not greater than the weight of X, $\vert\Gamma \vert\leq w(X)$.

Theorem 7. For any subset A in an ultrametric space (X,D) and any ultrametric d on A no greater than D|_A, there is an ultrametric $d^{\prime }$ on X that extends d. Moreover, there exists a greatest of such extensions. Corollary. There exists a push-out in the categories METR( c), METR$^{\circ }$, ULTRAMETR(c), and ULTRAMETR$^{\circ }$.