View source for Fundamental domain

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''of a discrete group  $  \Gamma $
of transformations of a topological space  $  X $''

A subset  $  D \subset  X $
containing elements from all the orbits (cf. [[Orbit|Orbit]]) of  $  \Gamma $, 
with exactly one element from orbits in general position. There are various versions of the exact definition of a fundamental domain. Sometimes a fundamental domain is any subset belonging to a given  $  \sigma $-
algebra (for example, the Borel  $  \sigma $-
algebra) and containing exactly one representative from each orbit. However, if  $  X $
is a topological manifold, then a fundamental domain is usually a subset  $  D \subset  X $
that is the closure of an open subset and is such that the subsets  $  \gamma D $, 
$  \gamma \in \Gamma $, 
have pairwise no common interior points and form a locally finite covering of  $  X $. 
For example, as a fundamental domain of the group of parallel translations of the plane  $  \mathbf R  ^ {2} $
by integer vectors one can take the square

$$ 
\{ {( x, y) \in \mathbf R  ^ {2} } : {
0 \leq  x \leq  1,\ 
0 \leq  y \leq  1 } \}
.
$$

The choice of a fundamental domain is, as a rule, non-unique.

====Comments====
The chambers of the [[Weyl group|Weyl group]]  $  W $
are examples of fundamental domains of  $  W $
in its reflection representation.