Several other metrization theorems follow as simple corollaries to Urysohn's Theorem.

Pavel Samuilovich Urysohn, Pavel Uryson () ( February 3, 1898, Odessa – August 17, 1924, Batz-sur-Mer ) was a Jewish mathematician who is best known for his contributions in the theory of dimension, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology.

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function.

Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function.

Unlike Urysohn's metrization theorem, which provides a sufficient condition for metrization, this theorem provides both a necessary and sufficient condition for a topological space to be metrizable.

Urysohn's lemma has led to the formulation of other topological properties such as the ' Tychonoff property ' and ' completely Hausdorff spaces '.

There are numerous characterizations that tell when a second countable topological space is metrizable, such as Urysohn's metrization theorem.

Urysohn's metrization theorem states that every second-countable, regular space is metrizable.

The theorem generalizes Urysohn's lemma and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal.

Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces.

A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmata, such as Bézout's lemma, Urysohn's lemma, Dehn's lemma, Euclid's lemma, Farkas ' lemma, Fatou's lemma, Gauss's lemma, Nakayama's lemma, Poincaré's lemma, Riesz's lemma, Schwarz's lemma, Itō's lemma and Zorn's lemma.

The first really useful metrization theorem was Urysohn's metrization theorem.

Urysohn's sister, Lina Neiman wrote a memoir about his life and childhood.

Illustration of Urysohn's " onion " function.

( See for instance Urysohn's lemma.

Several texts identify Tychonoff's theorem as the single most important result in general topology Willard, p. 120 ; others allow it to share this honor with Urysohn's lemma.

For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

It was then simplified in 1947, when Leon Henkin observed in his Ph. D. thesis that the hard part of the proof can be presented as the Model Existence Theorem ( published in 1949 ).

The Model Existence Theorem and its proof can somehow be formalized in the framework of PA.

In the General Possibility Theorem, Kenneth Arrow argues that if a legislative consensus can be reached through a simple majority, then minimum conditions must be satisfied, and these conditions must provide a superior ranking to any subset of alternative votes ( Arrow 1963 ).

This theorem was established by John von Neumann, who is quoted as saying " As far as I can see, there could be no theory of games … without that theorem … I thought there was nothing worth publishing until the Minimax Theorem was proved ".

More formally, we can state the Transfinite Recursion Theorem as follows.

Where the angle is a right angle, also known as the Hypotenuse-Leg ( HL ) postulate or the Right-angle-Hypotenuse-Side ( RHS ) condition, the third side can be calculated using the Pythagoras ' Theorem thus allowing the SSS postulate to be applied.

The play opens on 10 April 1809, in a garden front room of a country house in Derbyshire with tutor Septimus Hodge trying to distract his 13 year-old pupil Thomasina from her enquiries as to the meaning of a " carnal embrace " by challenging her to prove Fermat's Last Theorem so he can focus on reading the poem ' The Couch of Eros ', a piece written by another character, Mr. Ezra Chater.

The max-flow min-cut theorem is a special case of the duality theorem for linear programs and can be used to derive Menger's theorem and the König-Egerváry Theorem.

In social choice theory, Arrow ’ s impossibility theorem, the General Possibility Theorem, or Arrow ’ s paradox, states that, when voters have three or more distinct alternatives ( options ), no rank order voting system can convert the ranked preferences of individuals into a community-wide ( complete and transitive ) ranking while also meeting a specific set of criteria.

By the Chinese Remainder Theorem, each can further be decomposed into a direct sum of submodules of the form, where is a power of a prime ideal.

For any flow, you can write the equations of the flow in terms of vorticity rather than velocity by simply taking the curl of the flow equations that are framed in terms of velocity ( may have to apply the 2nd Fundamental Theorem of Calculus to do this rigorously ).

The local density of points in such systems obeys Liouville's Theorem, and so can be taken as constant.

Using Wilson's Theorem, for any odd prime we can rearrange the left hand side of

Moreover, as long as the polynomial factors at each stage are relatively prime ( which for polynomials means that they have no common roots ), one can construct a dual algorithm by reversing the process with the Chinese Remainder Theorem.

This was derived from the Second Law ( any of the two, Clausius ' or Lord Kelvin's statement can be used since they are equivalent ) and using the Clausius ' Theorem ( see Kerson Huang ISBN-13: 978-0471815181 ).

Bayes ' Theorem shows that the probability will never reach exactly 0 or 100 % ( no absolute certainty in either direction ), but it can still get very close to either extreme.

The Gauss – Bonnet Theorem can be seen as a special instance in the theory of characteristic classes.

The Second Fundamental Theorem can also be derived from the metric-topological theory of Ahlfors, which can be considered as an extension of the Riemann – Hurwitz formula to the coverings of infinite degree.

Many other Picard-type theorems can be derived from the Second Fundamental Theorem.

As another corollary from the Second Fundamental Theorem, one can obtain that

The classical Stokes's Theorem and Divergence Theorem can be seen as special cases of this.

* Theorem Let X be a normed space.

: Turing's thesis: " Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i. e. by one of his machines, is equivalent to Church's thesis by Theorem XXX.

: Theorem on projections: Let the function f: X → B be such that a ~ b → f ( a )

This result is now known as Church's Theorem or the Church – Turing Theorem ( not to be confused with the Church – Turing thesis ).

Fine, Do Correlations need to be explained ?, in Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem, edited by Cushing & McMullin ( University of Notre Dame Press, 1986 ).

Theorem: Let V be a topological vector space

By the Chinese Remainder Theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the only idempotents will be 0 and 1.

~ p ∨ p. Since p → p is true ( this is Theorem 2. 08, which is proved separately ), then ~ p ∨ p must be true.

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

Using Rogers ' characterization of acceptable programming systems, Rice's Theorem may essentially be generalized from Turing machines to most computer programming languages: there exists no automatic method that decides with generality non-trivial questions on the black-box behavior of computer programs.

On 13 August, 2012, this project was officially announced to be The Zero Theorem, set to start shooting in Bucharest on October 22, produced by Dean Zanuck ( son to the late Richard D. Zanuck who was to originally produce in 2009 ), worldwide sales handled by Voltage Pictures, Toronto and starring Academy Award winner Christoph Waltz in the lead, replacing Billy Bob Thornton who had been attached to the project in 2009.

On the other hand, the scholars invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of prime numbers into a physical question.

Theorem: The angle may be trisected if and only if is reducible over the field extension Q.