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 
.
