* Every preorder can be given a topology, the Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.

Every fifteen minutes they played a record ; the rest of the time it was commercials about a high-school correspondence course.

Every line is a set of points which can be put into a one-to-one correspondence with the real numbers.

Every correspondence prescription between phase space and Hilbert space, however, induces its own proper-product.

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

Every smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

Every topological group can be viewed as a uniform space in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.

* PASSIA Diaries and Annual Reports: Every year since 1988, PASSIA publishes its “ Diary ”, a unique annual resource book combining a comprehensive directory of contact information for Palestinian and interna tional institu tions operating in Palestine, a day-to-day calendar, and an agenda containing facts and figures, graphs, statistics, chronologies and maps related to Palestine and the Palestinians.

Every smooth manifold defined in this way has a natural diffeology, for which the plots correspond to the smooth maps from open subsets of R < sup > n </ sup > to the manifold.

Every rotation maps an orthonormal basis of R < sup > 3 </ sup > to another orthonormal basis.

Every covering map is a semicovering, but semicoverings satisfy the " 2 out of 3 " rule: given a composition of maps of spaces, if two of the maps are semicoverings, then so also is the third.

Every measurable entity maps into a cardinal function but not every cardinal function is the result of the mapping of a measurable entity.

Every station has detailed maps of the station and surrounding area showing the locations of each exit.

* Every topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.

* Every compact metric space is separable.

* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.

* Sequentially compact: Every sequence has a convergent subsequence.

* Countably compact: Every countable open cover has a finite subcover.

* Limit point compact: Every infinite subset has an accumulation point.

Every compact metric space is complete, though complete spaces need not be compact.

Every entire function can be represented as a power series that converges uniformly on compact sets.

* Every compact metric space ( or metrizable space ) is separable.

* Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.

Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.

Every continuous function on a compact set is uniformly continuous.

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article compact space.

* Every compact Hausdorff space of weight at most ( see Aleph number ) is the continuous image of ( this does not need the continuum hypothesis, but is less interesting in its absence ).

*( BCT2 ) Every locally compact Hausdorff space is a Baire space.

Every group has a presentation, and in fact many different presentations ; a presentation is often the most compact way of describing the structure of the group.

*( BCT2 ) Every locally compact Hausdorff space is a Baire space.

Every H * is very special in structure: it is pure-injective ( also called algebraically compact ), which says more or less that solving equations in H * is relatively straightforward.

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.

Every polyhedron, regular and irregular, convex and concave, has a dihedral angle at every edge.

Every antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides of an isosceles trapezoid.

Every nondegenerate triangle is strictly convex.

* Every locally convex topological vector space has a neighbourhood basis consisting of barrelled sets.

* Every metrisable locally convex space with continuous dual carries the Mackey topology, that is, or to put it more succinctly every Mackey space carries the Mackey topology

Every ( bounded ) convex polytope is the image of a simplex, as every point is a convex combination of the ( finitely many ) vertices.

Every convex centrally symmetric polyhedron P in R < sup > 3 </ sup > admits a pair of opposite ( antipodal ) points and a path of length L joining them and lying on the boundary ∂ P of P, satisfying

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

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

# Every infinite subset of X has a complete accumulation point.

# Every infinite subset of A has at least one limit point in A.

* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.

* Every separable metric space is homeomorphic to a subset of the Hilbert cube.

* Every separable metric space is isometric to a subset of the ( non-separable ) Banach space l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this is known as the Fréchet embedding.

* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

* Every separable metric space is isometric to a subset of the

Every element s, except a possible greatest element, has a unique successor ( next element ), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound.

Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.

* Every subset of Baire space or Cantor space is an open set in the usual topology on the space.

* Every arithmetical subset of Cantor space of < sup >( or?

Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable ( and therefore T4 ) and second countable.

It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.

* Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces.

* Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial.

* Every finite or cofinite subset of the natural numbers is computable.