* Any finite topological space, including the empty set, is compact.

* Any space carrying the cofinite topology is compact.

* Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification.

Note that for orientable compact surfaces without boundary, the Euler characteristic equals, where is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and counts the number of handles.

Concerned with the runaway success of the Ford Mustang, Chevrolet executives realized that their compact sporty car, the Corvair, would not be able to generate the sales volume of the Mustang due to its rear-engine design, as well as declining sales, partly due to the bad publicity from Ralph Nader's book, Unsafe at Any Speed.

Any compact orientable surface and any compact surface with non-empty boundary embeds in though any closed non-orientable surface needs.

Any bounded operator L that has finite rank is a compact operator ; indeed, the class of compact operators is a natural generalisation of the class of finite-rank operators in an infinite-dimensional setting.

* Any compact operator is strictly singular, but not vice-versa.

Any action of the group by continuous affine transformations on a compact convex subset of a ( separable ) locally convex topological vector space has a fixed point.

Any compact manifold has isoperimetric dimension 0.

( 2 ) Any sequence of continuous positive definite functions on G converging to 1 uniformly on compact subsets, converges to 1 uniformly on G.

Any compact Riemann surface of genus greater than 1 ( with its usual metric of constant curvature − 1 ) is a locally symmetric space but not a symmetric space.

* Any compact group is locally compact.

* Any discrete group is locally compact.

The canonical ring and therefore likewise the Kodaira dimension is a birational invariant: Any birational map between smooth compact complex manifolds induces an isomorphism between the respective canonical rings.

Any real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian.

Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system.

Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such form a subalgebra of the Lie Algebra of symplectic vector fields.

Any symplectic vector space admits a Darboux basis

Any symplectic manifold ( or indeed any almost symplectic manifold ) has a natural volume form.

Any symplectic manifold admits a compatible almost complex structure making it into an almost Kähler manifold.

# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.

* Any smooth manifold is a diffeological space.

Any Calabi – Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi – Yau manifold.

Any smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology.

# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.

Any covering space of a differentiable manifold may be equipped with a ( natural ) differentiable structure that turns p ( the covering map in question ) into a local diffeomorphism – a map with constant rank n.

Any vector bundle V over a manifold may be realized as the pullback of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle ; these universal Chern classes in turn can be explicitly written down in terms of Schubert cycles.

* Any open manifold admits a ( non-complete ) Riemannian metric of positive ( or negative ) curvature.

Any “ nice ” space such as a manifold or CW complex is semi-locally simply connected.

Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network.

* Any manifold with constant sectional curvature is an Einstein manifold — in particular:

Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy.

Any manifold is an n-handlebody, that is, any manifold is the union of handles.

Any handlebody decomposition of a manifold defines a CW complex decomposition of the manifold, since attaching an r-handle is the same, up to homotopy equivalence, as attaching an r-cell.

Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot – Carathéodory, defined as

Any timelike curve admits a comoving observer whose " time axis " corresponds to that curve, and, since no observer is privileged, we can always find a local coordinate system in which lightcones are inclined at 45 degrees to the time axis.

Any process involving virtual particles admits a schematic representation known as a Feynman diagram, which facilitates the understanding of calculations.

Any group scheme admits natural left and right actions on its underlying scheme by multiplication and conjugation.

* Any uniformly continuous function admits a minimal modulus of continuity, that is sometimes referred to as the ( optimal ) modulus of continuity of:

:: Theorem ( Lefschetz theorem on ( 1, 1 )- classes ) Any element of H < sup > 2 </ sup >( X, Z ) ∩ H < sup > 1, 1 </ sup >( X ) is the cohomology class of a divisor on X.