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

** CmptH, the category of all compact Hausdorff spaces

** Hausdorff topologies are better-behaved than those in arbitrary general topology.

** Hausdorff dimension

** The Banach – Tarski paradox.

** EPR paradox

** Shaar HaYichud-The Gate of Unity, Dovber Schneuri-A detailed explanation of the paradox of divine simplicity.

** Heat death paradox, a philosophical examination of the cosmological event

** Ehrenfest paradox and Ehrenfest theorem, named after him

** Eunectes murinus, the green anaconda, the largest species, is found east of the Andes in Colombia, Venezuela, the Guianas, Ecuador, Peru, Bolivia, Brazil and on the island of Trinidad.

** Eunectes notaeus, the yellow anaconda, a smaller species, is found in eastern Bolivia, southern Brazil, Paraguay and northeastern Argentina.

** Eunectes deschauenseei, the dark-spotted anaconda, is a rare species found in northeastern Brazil and coastal French Guiana.

** Eunectes beniensis, the Bolivian anaconda, the most recently defined species, is found in the Departments of Beni and Pando in Bolivia.

** Well-ordering theorem: Every set can be well-ordered.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.

** The Cartesian product of any family of nonempty sets is nonempty.

** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

** Every surjective function has a right inverse.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

** Antichain principle: Every partially ordered set has a maximal antichain.

** Every vector space has a basis.

** Every unital ring other than the trivial ring contains a maximal ideal.

** For every non-empty set S there is a binary operation defined on S that makes it a group.

** The closed unit ball of the dual of a normed vector space over the reals has an extreme point.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.

** In the product topology, the closure of a product of subsets is equal to the product of the closures.

** 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.

** Any union of countably many countable sets is itself countable.

* Hausdorff paradox

For various constructions of non-measurable sets, see Vitali set, Hausdorff paradox, and Banach – Tarski paradox.

The Hausdorff paradox and Banach – Tarski paradox show that you can take a three dimensional ball of radius 1, dissect it into 5 parts, move and rotate the parts and get two balls of radius 1.

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.

# REDIRECT Hausdorff paradox

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.

The prototypical example of a Banach algebra is, the space of ( complex-valued ) continuous functions on a locally compact ( Hausdorff ) space that vanish at infinity.

Equipped with the topology of pointwise convergence on A ( i. e., the topology induced by the weak -* topology of A < sup >∗</ sup >), the character space, Δ ( A ), is a Hausdorff compact space.

F is a constant and D is a parameter that Richardson found depended on the coastline approximated by L. He gave no theoretical explanation but Mandelbrot identified L with a non-integer form of the Hausdorff dimension, later the fractal dimension.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

* Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification.

Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.

Stone spaces, compact totally disconnected Hausdorff spaces, form the abstract framework in which these spectra are studied.

* The structure space of a commutative unital Banach algebra is a compact Hausdorff space.

* A compact subset of a Hausdorff space is closed.

This is to say, compact Hausdorff space is normal.

* Two compact Hausdorff spaces X < sub > 1 </ sub > and X < sub > 2 </ sub > are homeomorphic if and only if their rings of continuous real-valued functions C ( X < sub > 1 </ sub >) and C ( X < sub > 2 </ sub >) are isomorphic.

( Gelfand – Naimark theorem ) Properties of the Banach space of continuous functions on a compact Hausdorff space are central to abstract analysis.

* 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.

) In particular, every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.

* A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.

A Hausdorff space is H-closed if every open cover has a finite subfamily whose union is dense.

Embeddings into compact Hausdorff spaces may be of particular interest.

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.