A fundamental domain for a group action of 
by homeomorphisms on a topological space 
, under a common closed-domain convention, is a subset 
that is the closure
of its interior and whose translates cover 
,
 |
with 
whenever 
. Here 
denotes the interior
of 
, and 
is the image of 
under 
(Voight 2021, p. 631). Thus distinct translates of the
same fundamental domain have disjoint interiors, but
they may meet on their boundaries.
For example, a fundamental domain of the group of rotations by multiples of 
in 
is the closed upper half-plane 
. For rotations
by multiples of 
, the first quadrant,
including its boundary, is a fundamental domain.
The group 
acting on 
by translations has the fundamental
domains ![[0,1]](/images/equations/FundamentalDomain/Inline18.svg)
and ![[1/2,3/2]](/images/equations/FundamentalDomain/Inline19.svg)
. Their interiors
overlap. The nonoverlap condition concerns translates of a single chosen fundamental
domain, not two different fundamental domains. A closed fundamental domain may contain
multiple representatives of a group orbit on its boundary. Requiring exactly one representative of every
group orbit uses a different boundary convention.
This differs from a group block, whose translates are either identical or disjoint as sets. Translates of a closed fundamental domain may instead overlap on their boundaries.
