** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

# REDIRECT Hausdorff maximal principle

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.

es: Principio maximal de Hausdorff

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.

* Hausdorff maximal principle

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 maximal principle, a concept in mathematics

# REDIRECT Hausdorff maximal principle

The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma ( Kelley 1955: 33 ).

This states that every Hausdorff second-countable regular space is metrizable.

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.

* 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.

X is a Hausdorff space if any two distinct points of X can be separated by neighborhoods.

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.

* 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.

For example, if Y is a Hausdorff space, then ker f must be a closed set.

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.