If S is compact but not closed, then it has an accumulation point a not in S. Consider a collection consisting of an open neighborhood N ( x ) for each x ∈ S, chosen small enough to not intersect some neighborhood V < sub > x </ sub > of a.

Consider the open balls centered upon a common point, with any radius.

Consider for instance the open unit disc, a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss – Bonnet formula does not work.

Consider an open set on the real line and a function f defined on that set with real values.

Consider the open covering of which consists of:

Consider a scheme X and a cover by affine open subschemes Spec A < sub > i </ sub >.

Consider too, with what force, diligence and vivacity he has rendered back all this which in Johnson's neighbourhood, his " open sense " had so eagerly and freely taken in.

Consider the subset sum problem, an example of a problem that is easy to verify, but whose answer may be difficult to compute.

That means that some member of P ( S ), i. e., some subset of S, is not in the image of f. Consider the set:

Consider a function with its corresponding graph as a subset of the Cartesian product.

Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1 / n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε.

Consider the subset C of B, consisting of those B elements, which satisfy monic polynomial equations over A:

Consider the translates t + N of this subset.

Consider the set, subset of on which the multidimensional definite integral

Consider a subset and a mapping function

Geometric arrangement for Fresnel's calculation Consider the case of a point source located at a point P < sub > 0 </ sub >, vibrating at a frequency f. The disturbance may be described by a complex variable U < sub > 0 </ sub > known as the complex amplitude.

Suppose that U: D → C is a functor from a category D to a category C, and let X be an object of C. Consider the following dual ( opposite ) notions:

The Veterans Committee is the popular name of the National Baseball Hall of Fame Committee to Consider Managers, Umpires, Executives and Long-Retired Players, a committee of the U. S. Baseball Hall of Fame that provides an opportunity for Hall of Fame election to all individuals who are eligible for induction but ineligible for consideration by the Baseball Writers Association of America.

) Consider this conditional sentence: " If George W. Bush hadn't become president of the U. S. in 2001, Al Gore would have.

Consider that Air Canada has a major hub at Toronto that offers flights to several U. S. cities.

Consider that a perpetuity growth rate exceeding the annualized growth of the S & P 500 and / or the U. S. GDP implies that the company's cash flow will outpace and eventually absorb these rather large values.

In Consider Phlebas it is noted that Minds still find humans fascinating, especially their odd ability to sometimes achieve similarly advanced reasoning as their much more complex machine brains.

Consider a slightly more complex example:

Consider the complex Hilbert space L < sup > 2 </ sup > and the differential operator

Consider the complex Hilbert space L < sup > 2 </ sup >( R ), and the operator which multiplies a given function by x:

Consider the ( Euclidean ) complex plane equipped with the metric

Consider the Gram – Schmidt process applied to the columns of the full column rank matrix, with inner product ( or for the complex case ).

A fundamental part of ` Abdul-Bahá's teachings on evolution is the belief that all life came from the same origin: " the origin of all material life is one ..." He states that from this sole origin, the complete diversity of life was generated: " Consider the world of created beings, how varied and diverse they are in species, yet with one sole origin " He explains that a slow, gradual process led to the development of complex entities:

Consider the complex logarithm function log z.

* Consider the C *- algebra of complex square matrices.

Consider for example any compact connected complex manifold M: any holomorphic function on it is locally constant by Liouville's theorem.

Consider a complex scalar field φ, with the constraint that φ < sup >*</ sup > φ = v², a constant.

Consider a complex, real-world problem, like those of marketing or making policies for a nation, where there are many governing factors, and most of them cannot be expressed as numerical time series data, as one would like to have for building mathematical models.

Consider a once-punctured elliptic curve, given as the locus D of complex points satisfying, where and is a complex number.

Consider a d < sup > 6 </ sup > octahedral complex ( example IrBr < sub > 6 </ sub >< sup > 3 -</ sup >).

* Consider C, the field of complex numbers, as a 1-dimensional vector space.

Consider a polygon in the complex plane.

Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by Af.

Consider, for example, the implication this has for plane rotations.

Consider the plane spanned by and, where is a ket in the subspace perpendicular to.

Consider the special case in which the axis of rotation lies in the xy plane.

Consider two points A and B in two dimensional plane flow.

Consider two dimensional plane flow within a Cartesian coordinate system.

Consider a sphere B of radius 1 and a plane P touching B at the South Pole S of B.

Consider now the Minkowski plane: R < sup > 2 </ sup > equipped with the metric

* Consider a uniform layer of fluid over an infinite horizontal plane.

Consider a " small " light source located on-axis in the object plane of the lens.

Consider a plane with a compact arrangement of spheres on it.

Consider a pair of parallel lines in an affine plane A.

Consider the example of moving along a curve γ ( t ) in the Euclidean plane.

Consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of sides are arcs along each knot but is otherwise disjoint from the knots.

Consider a point in a continuum under a state of plane stress, or plane strain, with stress components and all other stress components equal to zero ( Figure 7. 1, Figure 8. 1 ).

Consider a set of points R ( R is a vector depicting a point in a Bravais lattice ) constituting a Bravais lattice, and a plane wave defined by:

Consider the illustration, depicting a plane intersecting a cone to form an ellipse ( the interior of the ellipse is colored light blue ).

Consider two proof masses vibrating in plane ( as in the MEMS gyro ) at frequency.

Consider region D in the plane: a unit circle or general polygon — the asymptotics of the problem, which are the interesting aspect, aren't dependent on the exact shape.

* Consider a triangle in the plane with unequal sides.

Consider an inertial observer in Minkowski spacetime who encounters a sandwich plane wave.

Consider a plane wave where all perturbed quantities vary as exp ( i ( kx-ωt )).