S.SANTHANAM

  Metric spaces [ edit ] Definition [ edit ] A point  �  of the  metric space   ( � , � )  is the  limit  of the  sequence   ( � � )  if: For each  real number   � > 0 , there is a  natural number   �  such that, for every natural number  � ≥ � , we have  � ( � � , � ) < � . Symbolically, this is: ∀ � > 0 ( ∃ � ∈ � ( ∀ � ∈ � ( � ≥ � ⟹ � ( � � , � ) < � ) ) ) . This coincides with the definition given for real numbers when  � = �  and  � ( � , � ) = | � − � | . Properties [ edit ] When it exists, the limit of a sequence is unique, as distinct points are separated by some positive distance, so for  �  less than half this distance, sequence terms cannot be within a distance  �  of both points. For any  continuous function   f , if  lim � → ∞ � �  exists, then  lim � → ∞ � ( � � ) = � ( lim � → ∞ � � ) . In fact, a  function ...

  Functions between metric spaces [ edit ] Euler diagram  of types of functions between metric spaces. Unlike in the case of topological spaces or algebraic structures such as  groups  or  rings , there is no single "right" type of  structure-preserving function  between metric spaces. Instead, one works with different types of functions depending on one's goals. Throughout this section, suppose that  ( � 1 , � 1 )  and  ( � 2 , � 2 )  are two metric spaces. The words "function" and "map" are used interchangeably. Isometries [ edit ] Main article:  Isometry One interpretation of a "structure-preserving" map is one that fully preserves the distance function: A function  � : � 1 → � 2  is  distance-preserving [12]  if for every pair of points  x  and  y  in  M 1 , � 2 ( � ( � ) , � ( � ) ) = � 1 ( � , � ) . It follows from the metric space axioms that a distance-preserving function is ...