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( A Hausdorff space or T < sub > 2 </ sub > space is a topological space in which any two distinct points are separated by neighbourhoods.
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Hausdorff and space
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.
* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).
* Uniform algebra: A Banach algebra that is a subalgebra of C ( X ) with the supremum norm and that contains the constants and separates the points of X ( which must be a compact Hausdorff space ).
For ƒ ∈ C ( X ) ( with a compact Hausdorff space X ), one sees that:
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.
* 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.
* The structure space of a commutative unital Banach algebra is a compact Hausdorff space.
By the same construction, every locally compact Hausdorff space X is a dense subspace of a compact Hausdorff space having at most one point more than X ..
* A compact subset of a Hausdorff space is closed.
This is to say, compact Hausdorff space is normal.
( 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.
Hausdorff and T
In topology and related branches of mathematics, a Hausdorff space, separated space or T < sub > 2 </ sub > space is a topological space in which distinct points have disjoint neighbourhoods.
Of the many separation axioms that can be imposed on a topological space, the " Hausdorff condition " ( T < sub > 2 </ sub >) is the most frequently used and discussed.
This condition is the third separation axiom ( after T < sub > 0 </ sub > and T < sub > 1 </ sub >), which is why Hausdorff spaces are also called T < sub > 2 </ sub > spaces.
X is a Tychonoff space, or T < sub > 3½ </ sub > space, or T < sub > π </ sub > space, or completely T < sub > 3 </ sub > space if it is both completely regular and Hausdorff.
* R. Brown, K. Hardie, H. Kamps, T. Porter: The homotopy double groupoid of a Hausdorff space., Theory Appl.
It follows that if a topological group is T < sub > 0 </ sub > ( Kolmogorov ) then it is already T < sub > 2 </ sub > ( Hausdorff ), even T < sub > 3½ </ sub > ( Tychonoff ).
A normal Hausdorff space is also called a T < sub > 4 </ sub > space.
These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T < sub > 5 </ sub > spaces, and perfectly normal Hausdorff spaces, or T < sub > 6 </ sub > spaces.
A T < sub > 4 </ sub > space is a T < sub > 1 </ sub > space X that is normal ; this is equivalent to X being Hausdorff and normal.
A completely T < sub > 4 </ sub > space, or T < sub > 5 </ sub > space is a completely normal Hausdorff topological space X ; equivalently, every subspace of X must be a T < sub > 4 </ sub > space.
The term " T < sub > 3 </ sub > space " usually means " a regular Hausdorff space ".
A T < sub > 3 </ sub > space or regular Hausdorff space is a topological space that is both regular and a Hausdorff space.
Hausdorff and <
* 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.
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.
The product is a Boolean space ( compact, Hausdorff and totally disconnected ), and X < sub > F </ sub > is a closed subset, hence again Boolean.
The following theorem represents positive linear functionals on C < sub > c </ sub >( X ), the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space X.
A ( topological ) surface is a nonempty second countable Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E < sup > 2 </ sup >.
More generally, a ( topological ) surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the upper half-plane H < sup > 2 </ sup >.
Since C is Hausdorff and separable it follows that C has cardinality equal to 2 < sup > ℵ < sub > 0 </ sub ></ sup > — the same cardinality as the cardinality of the reals.
Hausdorff and 2
Technically speaking, any surface in three-dimensional space has a topological dimension of 2, and therefore any fractal surface in three-dimensional space has a Hausdorff dimension between 2 and 3.
The sponge has a Hausdorff dimension of ( log 20 ) / ( log 3 ) ( approximately 2. 726833 ).
These conditions are separation axioms that are somewhat stronger than the more familiar Hausdorff axiom T < sub > 2 </ sub >.
Likewise, the two dimensional Hausdorff measure of a measurable subset of R < sup > 2 </ sup > is proportional to the area of the set.
Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers where 2 =
In fact a spectral space is T < sub > 1 </ sub > if an only if it is Hausdorff ( or T < sub > 2 </ sub >) if and only if it is a boolean space.
Any Hausdorff ( T < sub > 2 </ sub >) space is sober ( the only irreducible subsets being points ), and all
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.
In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2.
Mandelbrot shows how to calculate the Hausdorff dimension of each of these curves, each of which has a dimension D between 1 and 2 ( he also mentions but does not give a construction for the space-filling Peano curve, which has a dimension exactly 2 ).
For example, a regular space ( called T < sub > 3 </ sub >) does not have to be a Hausdorff space ( called T < sub > 2 </ sub >), at least not according to the simplest definition of regular spaces.
* A compact Hausdorff space X with < math >| X | < 2 ^ k </ math > is sequentially compact, i. e., every sequence has a convergent subsequence.
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