* 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 Lie group is parallelizable, and hence an orientable manifold ( there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity )

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 Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions.

Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if

Every Kähler manifold is also a symplectic manifold.

* Every Lie algebra is a Lie algebroid over the one point manifold.

* Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.

Every Hermitian manifold is a complex manifold which comes naturally equipped with a Hermitian form and an integrable, almost complex structure.

Every closed manifold is the boundary of the non-compact manifold.

Every closed manifold is such that, so for every.

Every complex manifold is itself an almost complex manifold.

Every hyperkähler manifold M has a 2-sphere of complex structures ( i. e. integrable almost complex structures ) with respect to which the metric is Kähler.

( Every Calabi – Yau manifold in 4 ( real ) dimensions is a hyperkähler manifold, because SU ( 2 ) is isomorphic to Sp ( 1 ).

* The embedding theorem for Stein manifolds states the following: Every Stein manifold of complex dimension can be embedded into by a biholomorphic proper map.

* Every Stein manifold is holomorphically spreadable, i. e. for every point, there are holomorphic functions defined on all of which form a local coordinate system when restricted to some open neighborhood of.

Every compact smooth manifold of dimension 2n, which has only handles of index ≤ n, has a Stein structure provided n > 2, and when n = 2 the same holds provided the 2-handles are attached with certain framings ( framing less than the Thurston-Bennequin framing ).

Every complete, connected, simply-connected manifold of constant negative curvature − 1 is isometric to the real hyperbolic space H < sup > n </ sup >.

Every manifold has an underlying topological manifold, obtained simply by forgetting the additional structure.

Every legislator from Brasstown Bald to Folkston is going to have his every vote subjected to the closest scrutiny as a test of his political allegiances, not his convictions.

Every detail in his interpretation has been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout.

Every taxpayer is well aware of the vast size of our annual defense budget and most of our readers also realize that a large portion of these expenditures go for military electronics.

Every single problem touched on thus far is related to good marketing planning.

Every few days, in the early morning, as the work progressed, twenty men would appear to push it ahead and to shift the plank foundation that distributed its weight widely on the Rotunda pavement, supported as it is by ancient brick vaulting.

Every dream, and this is true of a mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought, is a psychic phenomenon for which no explanation is available.

Every man in every one of these houses is a Night Rider.

Every library borrower, or at least those whose taste goes beyond the five-cent fiction rentals, knows what it is to hear the librarian say apologetically, `` I'm sorry, but we don't have that book.

Every community, if it is alive has a spirit, and that spirit is the center of its unity and identity.

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 natural-born citizen of a foreign state who is also an American citizen and every natural-born American citizen who is a citizen of a foreign land owes a double allegiance, one to the United States, and one to his homeland ( in the event of an immigrant becoming a citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming a citizen of another nation ).

Every line of written text is a mere reflection of references from any of a multitude of traditions, or, as Barthes puts it, " the text is a tissue of quotations drawn from the innumerable centres of culture "; it is never original.

Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

Every year, on the last Sunday in April, there is an ice fishing competition in the frozen estuarine waters of the Anadyr River's mouth.

Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.