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The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

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## Some Related Sentences

Hausdorff and maximal

**

__Hausdorff____maximal__**principle**: In**any****partially****ordered****set****,****every****totally****ordered****subset****is****contained****in****a**__maximal__**totally****ordered****subset****.**
In mathematics

**,**the__Hausdorff____maximal__**principle****is**an alternate and earlier formulation of Zorn's lemma proved by Felix__Hausdorff__**in**1914 ( Moore 1982: 168 ).**The**

__Hausdorff__

__maximal__

**principle**

**is**one of many statements equivalent to the axiom of choice over Zermelo – Fraenkel

**set**theory

**.**

**The**proof

**that**the

__Hausdorff__

__maximal__

**principle**

**is**equivalent to Zorn's lemma

**is**very similar to this proof

**.**

Spec ( R )

**is****a**compact space**,**but almost never__Hausdorff__:**in**fact**,**the__maximal__ideals**in**R are precisely the closed points**in**this topology**.**
Under this duality

**,****every**compact__Hausdorff__space**is**associated with the algebra of continuous complex-valued functions on**,**and**every**commutative C *- algebra**is**associated with the space of its__maximal__ideals**.**

Hausdorff and principle

**The**

__principle__

**is**also called the

__Hausdorff__maximality theorem or the Kuratowski lemma ( Kelley 1955: 33 ).

Hausdorff and states

It

__states__**that****a**topological space**is**metrizable if and only if it**is**regular**,**__Hausdorff__and has**a**σ-locally finite base**.**
* Open mapping theorem ( topological groups )

__states__**that****a**surjective continuous homomorphism of**a**locally compact__Hausdorff__group G onto**a**locally compact__Hausdorff__group H**is**an open mapping if G**is**σ-compact**.****The**equicontinuity appears

**in**the formulation of Ascoli's theorem

**,**which

__states__

**that**

**a**

**subset**of C ( X ), the space of continuous functions on

**a**compact

__Hausdorff__space X

**,**

**is**compact if and only if it

**is**closed

**,**pointwise bounded and equicontinuous

**.**

*

**The**strong Whitney embedding theorem__states__**that****any**smooth m-dimensional manifold ( required also to be__Hausdorff__and second-countable ) can be smoothly embedded**in**Euclidean 2m-space**,**if m > 0**.**
In mathematics

**,**the__Hausdorff__paradox**,**named after Felix__Hausdorff__**,**__states__**that**if you remove**a**certain countable**subset**of the sphere S < sup > 2 </ sup >, the remainder can be divided into three disjoint subsets A**,**B and C such**that**A**,**B**,**C and B ∪ C are all congruent**.****The**theorem

__states__

**that**

**a**topological space

**is**metrizable if and only if it

**is**regular and

__Hausdorff__and has

**a**countably locally finite ( i

**.**e., σ-locally finite ) basis

**.**

Hausdorff and any

*

**The**spectrum of__any__commutative ring with the Zariski topology (**that****is****,**the**set**of all prime ideals )**is**compact**,**but never__Hausdorff__( except**in**trivial cases ).
Since

**every**compact__Hausdorff__space**is****a**Tychonoff space**,**and**every**subspace of**a**Tychonoff space**is**Tychonoff**,**we conclude**that**__any__space possessing**a**__Hausdorff__compactification must be**a**Tychonoff space**.**
In mathematics

**,**the__Hausdorff__dimension ( also known as the__Hausdorff__– Besicovitch dimension )**is**an extended non-negative real number associated with__any__metric space**.**
A first countable

**,**separable__Hausdorff__space (**in**particular**,****a**separable metric space ) has at most the continuum cardinality c**.**In such**a**space**,**closure**is**determined by limits of sequences and__any__sequence has at most one limit**,**so there**is****a**surjective map from the**set**of convergent sequences with values**in**the countable dense**subset**to the points of X**.**
Given

__any__embedding of**a**Tychonoff space X**in****a**compact__Hausdorff__space K the closure of the image of X**in**K**is****a**compactification of X**.**
It

**is**characterised by the universal property**that****,**given**a**continuous map f from X to__any__other compact__Hausdorff__space Y**,**there**is****a**unique continuous map g from βX to Y**that**extends f**in**the sense**that**f**is**the composition of g and j**.**
This

**is**an example of the following fact:__any__**subset**of R of__Hausdorff__dimension strictly greater than zero must be uncountable**.**
In the end

**,**this**is**not**a**serious restriction —__any__topological group can be made__Hausdorff__**in****a**canonical fashion**.****The**Stone – Čech compactification βX of

**a**topological space X

**is**the largest compact

__Hausdorff__space " generated " by X

**,**

**in**the sense

**that**

__any__map from X to

**a**compact

__Hausdorff__space factors through

βX

**is****a**compact__Hausdorff__space together with**a**continuous map from X and has the following universal property:__any__continuous map f: X → K**,**where K**is****a**compact__Hausdorff__space**,**lifts uniquely to**a**continuous map βf: βX → K**.****The**special property of the unit interval needed for this construction to work

**is**

**that**it

**is**

**a**cogenerator of the category of compact

__Hausdorff__spaces: this means

**that**if f and g are

__any__two distinct maps from compact

__Hausdorff__spaces A to B

**,**then there

**is**

**a**map h from B to such

**that**hf and hg are distinct

**.**

In the case where

**is**locally compact**,**e**.**g**.**or**,**the image of forms an open**subset**of**,**or indeed of__any__compactification**,**( this**is**also**a**necessary condition**,**as an open**subset**of**a**compact__Hausdorff__space**is**locally compact ).
( A

__Hausdorff__space or T < sub > 2 </ sub > space**is****a**topological space**in**which__any__two distinct points are separated by neighbourhoods**.**
Although

__Hausdorff__spaces aren't generally regular**,****a**__Hausdorff__space**that****is**also ( say ) locally compact will be regular**,**because__any____Hausdorff__space**is**preregular**.**
As described above

**,**__any__completely regular space**is**regular**,**and__any__T < sub > 0 </ sub > space**that****is**not__Hausdorff__( and hence not preregular ) cannot be regular**.**

Hausdorff and set

* 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**.**
Felix

__Hausdorff__( November 8**,**1868 – January 26**,**1942 ) was**a**German mathematician who**is**considered to be one of the founders of modern topology and who contributed significantly to__set__theory**,**descriptive__set__theory**,**measure theory**,**function theory**,**and functional analysis**.**
For

**a**" well-behaved "__set__X**,**the__Hausdorff__dimension**is**the unique number d such**that**N ( r ) grows as 1 / r < sup > d </ sup > as r approaches zero**.**
As evidence

**,**she quotes Felix__Hausdorff__: " A__set__**is**formed by the grouping together of single objects into**a**whole**.****The**Cantor

__set__

**is**an uncountable

**subset**of R

**.**

**The**Cantor

__set__

**is**

**a**fractal and has

__Hausdorff__dimension greater than zero but less than one ( R has dimension one ).

where the product

**is**over all maps from X to compact__Hausdorff__spaces C**.**This works intuitively but fails for the technical reason**that**the collection of all such maps**is****a**proper class rather than**a**__set__**.**
There are several ways to modify this idea to make it work ; for example

**,**one can restrict the compact__Hausdorff__spaces C to have underlying__set__P ( P ( X )) ( the power__set__of the power__set__of X ), which**is**sufficiently large**that**it has cardinality at least equal to**that**of**every**compact__Hausdorff____set__to which X can be mapped with dense image**.**
This

**is****a**critical failure: if X**is****a**completely regular__Hausdorff__space**,**there**is****a**natural embedding from X into < sup > C ( X ,)</ sup >, where C ( X ,)**is**the__set__of continuous maps from X to**.**
* It

**is****in**general not true**that**the closure of**every**open__set__**is**open**,**i**.**e**.**not**every****totally**disconnected__Hausdorff__space**is**extremally disconnected**.**
Conversely

**,**if X**is****a**__Hausdorff__space and ker f**is****a**closed__set__**,**then the coimage of f**,**if given the quotient space topology**,**must also be**a**__Hausdorff__space**.**0.161 seconds.