The term hyperspace has distinct meanings in geometry, linear algebra, and topology.
In geometric usage, a hyperspace is a space having dimension 
.
In linear algebra, a hyperspace of a vector space 
is a subspace 
of codimension 1, also called
a linear hyperplane (Hoffman and Kunze 1971, pp. 101,
109-110). If 
has finite dimension 
,
then 
has dimension 
. Such a subspace is
the linear transformation kernel of
a nonzero linear functional,
which is determined by 
only up to multiplication by a nonzero scalar.
In topology, a hyperspace is a topological space whose points are specified subsets
of another topological space 
.
For example, if 
is a compact metric
space, the collection of its nonempty closed sets,
conventionally denoted 
in hyperspace theory, can be equipped with the Hausdorff
metric. These closed sets are compact
sets. Another important hyperspace is 
, the collection of nonempty
closed sets in 
that are also connected sets,
with the same Hausdorff metric (Illanes and Nadler
1999, Chap. 1). Thus a point of such a hyperspace
represents an entire subset of 
,
rather than a point of 
.
The regular subcontinua of a metric continuum 
form the hyperspace 
,
consisting of the subcontinua equal to the topological
closure of their interior in 
(Ordoñez 2018).
For an arbitrary topological space 
,
the family of nonempty compact sets in 
can instead be made into a hyperspace using the Vietoris
topology.
