文档介绍：5 Brouwer’s Fixed Point Theorem and the Jordan Curve
Theorem
2
In this section we use the norm |x| = max{|x1|, |x2|} for x = (x1, x2) ∈ R and take as the unit
circle the set S1 = {x ∈ R2 / |x| = 1} with the usual topology inherited from R2. Thus S1 is
the boundary of I2, where I = [0, 1] with the usual topology inherited from R, so it is actually
in the shape of a square which is homeomorphic to our usual concept of a circle. For a function
2
f whose range is in R we denote by fi the ith coordinate function of f.
Definition A hex board is an array of regular hexagons arranged into a diamond shape in
such a way that there is the same number of hexagons along each side of the board.
The game of hex is played on a hex board in the following manner. Two players alternate
and the aim of the game is to connect pairs of opposite sides. More precisely, if the players are
labelled × and ◦ then one pair of opposite sides is labelled × and the other ◦. In turn each player
labels a previously unlabelled hexagon with her or his symbol. The winner is the player who first
obtains a connected path of adjacent hexagons stretching between the sides of that player’s label.
The picture below gives an example of a hex board in which there are six hexagons along
each side.
T
T
T T
T T
T T T
T T T
T T T T
T T T T
T