I am currently studying for my Analysis exam and ran into problems while working on an old exam. I hope that some one could help me with it.
The following two tables show the host cities and the corresponding European or World Champions of some past European or World Championships in soccer. (To not overload it, only the largest host country is listed, when a championship was played in more than one country).
| Year | Host | Europe master |
|---|---|---|
| 1964 | Spain | Spain |
| 1980 | Belgium | Germany |
| 1980 | Italy | Germany |
| 1984 | France | France |
| 1988 | Germany | Netherlands |
| 1996 | England | Germany |
| 2000 | Netherlands | France |
| 2012 | Ukraine | Spain |
| Year | Host | World master |
|---|---|---|
| 1970 | Mexico | Brasilien |
| 1974 | Germany | Germany |
| 1982 | Spain | Italy |
| 1990 | Italy | Germany |
| 1994 | USA | Brasiel |
| 2002 | Japan | Brasiel |
| 2010 | Southafrica | Spain |
| 2014 | Brasilien | Germany |
Consider for the sets $H_{1}$ and $H_{2}$ if there exist a metric, so that $H_{1}$ or $H_{2}$ are complete metric spaces and the mapping that assigns a host country to the corresponding europe or world champion country is a contraction.
$H_{1}:=\{Spain, Belgium, Italy, France, Germany, England, Netherlands, Ukraine\}$
or
$H_{2}:=\{Mexico, Germany, Spain, Italy, USA, Japan, Southafrica, Brasilien \}$
My idea:
The definition of a metric on a set is a function.
$$d:X \times X \rightarrow [0,\infty)$$ where the three axioms of the metric are fullfiled, but how do I proceed from the definition. I'm really at a loss here.
