In fact, every compact metric space is a continuous image of the Cantor set.

The utility of Cauchy sequences lies in the fact that in a complete metric space ( one where all such sequences are known to converge to a limit ), the criterion for convergence depends only on the terms of the sequence itself.

The Euclidean structure makes E < sup > n </ sup > an inner product space ( in fact a Hilbert space ), a normed vector space, and a metric space.

Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian ( that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish ).

In fact, the notion of " metric " is a generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance.

What distinguishes these tasks from other potential and actual NLP tasks is not only the volume of research devoted to them but the fact that for each one there is typically a well-defined problem setting, a standard metric for evaluating the task, standard corpora on which the task can be evaluated, and competitions devoted to the specific task.

The impetus arose from the fact that complete analytical solutions for the metric of a covariant N-body system have proven elusive in General Relativity.

One of the earliest was a competition to win ' a ton of money ' a pointed satire of tabloid newspapers promising huge cash prizes to boost circulation-the prize was in fact a metric tonne of one-and two-pence pieces, equivalent to a few hundred pounds sterling.

Significant problems with the metric of this form stem from the fact that all known warp drive spacetimes violate various energy conditions.

In fact, this scaling factor was introduced as a simple means to facilitate backward compatility between EIGRP and IGRP: In IGRP, the overall metric is a 24-bit value while EIGRP uses a 32-bit value to express this metric.

In fact, the space of all Lipschitz functions on a compact metric space is dense in the Banach space of continuous functions, an elementary consequence of the Stone – Weierstrass theorem.

The proofs of BCT1 and BCT2 for arbitrary complete metric spaces require some form of the axiom of choice ; and in fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.

In fact, the Heine – Borel theorem for arbitrary metric spaces reads:

That J and Ω usually have the same coordinate expression ( up to an overall sign ) is simply a consequence of the fact that the metric g is usually the identity matrix.

Abelian surfaces have a Ricci flat metric with holonomy strictly smaller than SU ( 2 ) ( in fact trivial ) so are not Calabi – Yau manifolds according to such definitions.

The fact that an animal weighing over 4. 5 metric tons ( 5 short tons ) could have a brain of no more than 80 grams ( 2. 8 oz ) contributed to the popular old idea that dinosaurs were unintelligent, an idea now largely rejected.

However this minimal decomposition is not necessarily the one produced by Ricci flow ; if fact, the Ricci flow can cut up a manifold into geometric pieces in many inequivalent ways, depending on the choice of initial metric.

This quadratic form is interpreted in physics as the metric tensor of Minkowski spacetime, so this definition is simply a restatement of the fact that Lorentz transformations are precisely the linear transformations which are also isometries of Minkowski spacetime.

In fact, this connection is the Levi-Civita connection for the metric on S < sup > 2 </ sup > inherited from R < sup > 3 </ sup >.

This fact motivates, for instance, the introduction of the Ricci flow equation as a natural extension of the heat equation for the metric.

In fact, by taking the Taylor expansion of the metric applied to a Jacobi field along a radial geodesic in the normal coordinate system, one has

For the study of the mechanical behavior of solids and fluids these are assumed to be continuous bodies, which means that the matter fills the entire region of space it occupies, despite the fact that matter is made of atoms, has voids, and is discrete.

This is not true for infinite dimensions ; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.

In fact, the converse is also true ; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.

For example, the conservation of energy follows from the time-invariance of physical systems, and the fact that physical systems behave the same regardless of how they are oriented in space gives rise to the conservation of angular momentum.

As a result of their rearing, Kal-El grows up to think of himself as Clark Kent, completely unaware of his alien heritage until he was well into adulthood ( the fact that the Kents found him sometime just after the Cold War meant that the Kents assumed the rocket they found him in was from an Earth-based space program-possibly that of Communist Russia-and that Clark had superhuman abilities because he was either a metahuman or some kind of genetic experiment ).

Sales were, unsurprisingly, very strong, and helped by the fact that several competitors had just entered the market with machines aimed directly at the PDP-5's market space, which the PDP-8 trounced.

To support this model, a version of Piaget ’ s Transitive Inference Task was used to show that the hippocampus is in fact used as the memory space.

In fact, every real n-dimensional vector space V is isomorphic to R < sup > n </ sup >.

In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.

The solution of the Schrödinger equation ( wave equations ) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic ( it is radially symmetric in space and only depends on the distance to the nucleus ).

In fact, the projections of a vector space are exactly the idempotent elements of the ring of linear transformations of the vector space.

Because the existence of these points of space as such, that is, as truly indivisible unities, has its basis in the fact that the points are one with the bodies that fill them ; because, therefore, all possible space is coincident with the physical world ( as in Wycliffe's system, in general, reality and possibility correspond ), there can as little be a vacuum as bounding surfaces that are common to different bodies.

In nearly two-thirds of American mosques, women pray behind partitions or in separate areas, not in the main prayer hall ; some mosques do not admit women at all due to the lack of space and the fact that some prayers, such as the Friday Jummah, are mandatory for men but optional for women.

The broadcast was entitled " Suns, Spaceships and Bug-Eyed Monsters "-a light-hearted look at how science fiction had become science fact, as well as how ideas of space travel had become reality through the 20th century.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

In fact, one important aspect of their very religion is the annihilation of men ''.

In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.

The fact is due mainly to international wars, both hot and cold.

While the pattern is uneven, some having gained more than others, nationalism has in fact served the Western peoples well.

In point of fact, this is a beige box with a bright red door, about one and a half feet square and hung from the wall about six feet from the door to Wisman's right.

The fact is that the Southern Confederacy differed from the earlier one almost as much as the Federal Constitution did.

To my knowledge, Lincoln remains the only Head of State and Commander-in-Chief who, while fighting a fearful war whose issue was in doubt, proved man enough to say this publicly -- to give his foe the benefit of the fact that in all human truth there is some error, and in all our error, some truth.

In fact the accumulation of the hardware of destruction is day by day increasing our fear of each other.

The new fact the initiates of this cult have to learn is that they must move toward simplicity.

The magic circle is, in fact, a symbol of and preparation for the metaphysical orgasm ''.

Operating as a one man police force in fact if not in name, he is at once more independent and more dedicated than the police themselves.

`` I may possibly be a greater risk than is the normal person of my age '', the President had said on February 29th of the election year, ignoring the fact that no one of his age had ever lived out another term.

In the incessant struggle with recalcitrant political fact he learns to focus the essence of a problem in the significant detail, and to articulate the distinctions which clarify the detail as significant, with what is sometimes astounding rapidity.

we accord it its place there, and in Lawrence's treatment we are given the innocent fantasy of a child, in fact, the form in which oedipal love is expressed in childhood.

There is probably some significance in the fact that two of the best incest stories I have encountered in recent years are burlesques of the incest myth.

How much they esteemed him is shown by the fact that their underground committee selected him as one of the few who would be helped to escape.

That is not to deny that he has been aware of traditions, of course, that he is steeped in them, in fact, or that he has dealt with them, in his books.

There is evidence to suggest, in fact, that many authors of the humorous sketches were prompted to write them -- or to make them as indelicate as they are -- by way of protesting against the artificial refinements which had come to dominate the polite letters of the South.

It seems quite obvious that all the really difficult tasks of human beings arise from the fact that man is not one, but many.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

Perhaps the mere fact that by plucking on the nerves nature can awaken in the most ordinary of us, temporarily anyway, the sleeping poet, and in poets can discover their immortality, is the most remarkable of all the remarkable phenomena to which we can attest??