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** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.

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## Some Related Sentences

** and Baire

** and category

__**__In

**the**

__category__of Riemann surfaces

**,**an automorphism is a bijective biholomorphic map ( also called a conformal map ), from a surface to itself

**.**

__**__Systems Biology comes under this

__category__including reaction fluxes

**and**variable concentrations of metabolites

__**__Chamois leather also falls into

**the**

__category__of aldehyde tanning

**and**like brain tanning produces a highly water absorbent leather

**.**

__**__

**the**materials are subject to a subpoena-although many practitioners regard that fact

**as**a

__category__of permissible disclosure

**,**not

**as**a categorical exclusion from confidentiality ( because court-ordered secrecy provisions may apply even in case of a subpoena ).

__**__Hondo-Louis L ' Amour ( Note: Originally announced on February 15

**,**1954

**as**a nominee in this

__category__

**.**

__**__Poultry –

__category__of domesticated birds kept by humans for

**the**purpose of collecting their eggs

**,**or killing for their meat

**and**/ or feathers

**.**

** and theorem

__**__Tarski's

__theorem__: For every infinite set A

**,**there is a bijective map between

**the**sets A

**and**A × A

**.**

__**__König's

__theorem__: Colloquially

**,**

**the**sum of a sequence of cardinals is strictly less than

**the**product of a sequence of larger cardinals

**.**

__**__If S is a set of sentences of first-order logic

**and**B is a consistent subset of S

**,**then B is included in a set that is maximal among consistent subsets of S

**.**

**The**special case where S is

**the**set of all first-order sentences in a given signature is weaker

**,**equivalent to

**the**Boolean prime ideal

__theorem__; see

**the**section " Weaker forms " below

**.**

__**__

**The**Vitali

__theorem__on

**the**existence of non-measurable sets which states that there is a subset of

**the**real numbers that is not Lebesgue measurable

**.**

__**__Gödel's completeness

__theorem__for first-order logic: every consistent set of first-order sentences has a completion

**.**

__**__

**The**numbers

**and**are not algebraic numbers ( see

**the**Lindemann – Weierstrass

__theorem__); hence they are transcendental

**.**

__**__More generally

**,**Rademacher's

__theorem__extends

**the**differentiability result to Lipschitz mappings between Euclidean

**spaces**: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an

**open**set in R < sup > n </ sup >, is almost everywhere differentiable

**.**

__**__Artin reciprocity law

**,**a general

__theorem__in number theory that provided a partial solution to Hilbert's ninth problem

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