For any subset A of Euclidean space R < sup > n </ sup >, the following are equivalent:

The resulting compactification can be thought of as a circle ( which is compact as a closed and bounded subset of the Euclidean plane ).

The Euclidean minimum spanning tree of a set of points is a subset of the Delaunay triangulation of the same points, and this can be exploited to compute it efficiently.

In Euclidean space R < sup > n </ sup >, or any convex subset of R < sup > n </ sup >, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element.

A parametric model is a collection of distributions, each of which is indexed by a unique finite-dimensional parameter:, where is a parameter and is the feasible region of parameters, which is a subset of d-dimensional Euclidean space.

Similarly the set of all vectors in which is rational for all i is a countable dense subset of ; so for every the-dimensional Euclidean space is separable.

A ( topological ) surface is a nonempty second countable Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E < sup > 2 </ sup >.

For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S ( this point may be x itself ).

For instance, the general linear group GL ( n, R ) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL ( n, R ) as a subset of Euclidean space R < sup > n × n </ sup >.

A chart for a topological space M is a homeomorphism from an open subset U of M to an open subset of Euclidean space.

* A subset of Euclidean space R < sup > n </ sup > is compact if and only if it is closed and bounded.

A subset S of R < sup > n </ sup > is bounded with respect to the Euclidean distance if and only if it bounded as subset of R < sup > n </ sup > with the product order.

However, S may be bounded as subset of R < sup > n </ sup > with the lexicographical order, but not with respect to the Euclidean distance.

Typically, A is some subset of the Euclidean space R < sup > n </ sup >, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy.

If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open set centered at x which is contained in S.

In general, any subset of a Euclidean space R < sup > n </ sup > that is defined by a system of homogeneous linear equations will yield a subspace.

For a subset S of Euclidean space R < sup > n </ sup >, the following two statements are equivalent:

In vector calculus, a vector field is an assignment of a vector to each point in a subset of Euclidean space.

In geometry, a hyperplane of an n-dimensional space V is a " flat " subset of dimension n − 1, or equivalently, of codimension 1 in V ; it may therefore be referred to as an ( n − 1 )- flat of V. The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly ; in all cases however, any hyperplane can be given in coordinates as the solution of a single ( due to the " codimension 1 " constraint ) algebraic equation of degree 1 ( due to the " flat " constraint ).

In terms closer to those that Hilbert would have used, near the identity element e of the group G in question, we have some open set U in Euclidean space containing e, and on some open subset V of U we have a continuous mapping

Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.

The unitary group U ( n ) is endowed with the relative topology as a subset of M < sub > n </ sub >( C ), the set of all n × n complex matrices, which is itself homeomorphic to a 2n < sup > 2 </ sup >- dimensional Euclidean space.

** Every infinite game in which is a Borel subset of Baire space is determined.

* Cone ( linear algebra ), a subset of vector space closed under positive scaling

* Convex cone, a subset C of a vector space V is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C

A subset K of a topological space X is called compact if it is compact in the induced topology.

* The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some bounded linear operator.

For instance, a diagonal operator on the Hilbert space may have any compact nonempty subset of C as spectrum.

* A closed subset of a compact space is compact.

Note that, in a metric space, every compact subset is closed and bounded.

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

* A compact subset of a Hausdorff space is closed.

A halting probability can be interpreted as the measure of a certain subset of Cantor space under the usual probability measure on Cantor space.

Each of these strings p < sub > i </ sub > determines a subset S < sub > i </ sub > of Cantor space ; the set S < sub > i </ sub > contains all sequences in cantor space that begin with p < sub > i </ sub >.

Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.

An embedding of a topological space X as a dense subset of a compact space is called a compactification of X.

In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

Conversely, a complete subset of a metric space is closed.

The diameter of a subset of a metric space is the least upper bound of the distances between pairs of points in the subset.

The flow takes points of a subset A into the points Φ < sup > t </ sub >( A ) and invariance of the phase space means that

In particular, no subset of the points on the real line gets arbitrarily close to any real number.

According to Dumezil it falls into a particular subset of celestial gods, referred to in histories of religion as frame gods.

Although the words faith and belief are sometimes erroneously conflated and used as synonyms, faith properly refers to a particular type ( or subset ) of belief, as defined above.

An event is defined as a particular subset of the sample space to be considered.

A particular subset of fricatives are the sibilants.

Then is a totally ordered subset of A, hence there exists a maximal totally ordered subset containing, in particular A contains a maximal totally ordered subset.

In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.

A first countable, separable Hausdorff space ( in particular, a separable metric space ) has at most the continuum cardinality c. In such a space, closure is determined by limits of sequences and any sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of X.

* A subset of a language used for a particular purpose or in a particular social setting.

There are currently 20, 000 speakers of a subset of the Dutch dialect West Flemish in the arrondissement of Dunkirk and this particular subset is in danger of extinction within decades.

In particular, the convex hull of a subset of size m + 1 ( of the n + 1 defining points ) is an m-simplex, called an m-face of the n-simplex.

It can be shown as a consequence of the above properties that μ ( U ) > 0 for every non-empty open subset U. In particular, if G is compact then μ ( G ) is finite and positive, so we can uniquely specify a left Haar measure on G by adding the normalization condition μ ( G ) = 1.

This result matched the overall goal of whole language instruction and supported the use of phonics for a particular subset of reading skills, especially in the earliest stages of reading instruction.

They can oversee the selection of particular designated portions of a master document to produce different versions of the same document ( such as a " tutorial " or a " quick-reference guide ," where both of these consist of a subset of the material ).

In particular, any subset of the x < sub > i </ sub > has a marginal distribution that is also multivariate normal.

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

In particular the set of all real-valued Lipschitz functions on a compact metric space X having Lipschitz constant ≤ K is a locally compact convex subset of the Banach space C ( X ).

* In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. In the particular case of Hilbert spaces ( or more generally, inner product spaces ), an orthonormal basis is a set that is both complete and orthonormal.