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maximal fuzzy topologies
maximal fuzzy topologies

... A fuzzy topological space (X, T) is defined to be fuzzy connected [6] if it has no proper fuzzy clopen set. We define such a fuzzy connected space to be maximal fuzzy connected if any fuzzy connected topology stronger than T necessarily coincides with T. If A is a fuzzy set in a fuzzy topological sp ...
THE GEOMETRIES OF 3
THE GEOMETRIES OF 3

... is not homeomorphic to S 3 or U3. (Note that E3 and H3 are each homeomorphic to R3.) However, S2 x U and S2xSi each possesses a very natural metric which is simply the product of the standard metrics. This metric on S2 x U differs from those on S3, E3 and H3 by being anisotropic. If you stand at a p ...
ISOSPECTRAL AND ISOSCATTERING MANIFOLDS: A SURVEY
ISOSPECTRAL AND ISOSCATTERING MANIFOLDS: A SURVEY

... If (M, g) is a compact manifold with boundary and Dirichlet or Neumann conditions are imposed, the spectrum is again an infinite sequence of eigenvalues. Find manifolds (M1 , g1 ) and (M2 , g2 ) with the same spectrum. We will call closed manifolds with the same spectrum (including multiplicities) i ...
Metric geometry of locally compact groups
Metric geometry of locally compact groups

... additional properties. We often write discrete group for group, in view of later sections about topological groups, and especially locally compact groups. In the discrete setting, we distinguish four classes, each class properly containing the next one: (all) all discrete groups; (ct) countable grou ...
Metric geometry of locally compact groups
Metric geometry of locally compact groups

... [Mack–57]; see also Appendix B in [GlTW–05]. Gelfand and Raikov (1943) showed that LC-groups have “sufficiently many” irreducible continuous unitary representations [Dixm–69, Corollary 13.6.6]; this does not carry over to topological groups (examples of topological groups that are abelian, locally hom ...
Locally normal subgroups of totally disconnected groups. Part II
Locally normal subgroups of totally disconnected groups. Part II

... said to be h.j.i. if every non-trivial closed locally normal subgroup is open. • atomic type: |LN (G)| > 2 but LC(G) = {0, ∞}, there is a unique least element of LN (G) r {0}, the action of G on LN (G) is trivial and G is not abstractly simple. • non-principal filter type (abbreviated by NPF type): ...
On characterizations of Euclidean spaces
On characterizations of Euclidean spaces

... arbitrarily chosen unit) of the corresponding sector of the unit circle (normalized to 2π). This also defines an angular bisector. ...
Factorization homology of stratified spaces
Factorization homology of stratified spaces

... In the present work, we lay the foundations for a general theory of factorization homology, following the outline of [AF1] and after the originating work of Lurie in [Lu2]. We do this for stratified spaces and, more generally, B-manifolds, where B is a collection of basic singularity types endowed ...
What is a generic point? - Emory Math/CS Department
What is a generic point? - Emory Math/CS Department

... and every open subset of an irreducible scheme contains that scheme’s unique generic point. All of this material is standard, and [Liu] is a great reference. Let X be a scheme. Recall X is irreducible if its underlying topological space is irreducible. A (nonempty) topological space is irreducible i ...
Todd Quinto - Tufts Math Department
Todd Quinto - Tufts Math Department

Zero-pointed manifolds
Zero-pointed manifolds

... from Mfldn . For the formulation of duality results, what is the common geometric home for these two concepts? As one answer to this question, in Section 1 we define zero-pointed manifolds. Our category ZMfldn consists of pointed topological spaces M∗ for which the complement M := M∗ r ∗ is an n-man ...
Topological properties
Topological properties

... Definition 4.10. Let (X, T ) be a topological space. A connected component of X is any maximal connected subset of X, i.e. any connected C ⊂ X with the property that, if C ′ ⊂ X is connected and contains C, then C ′ must coincide with C. Proposition 4.11. Let (X, T ) be a topological space. Then (i) ...
Affine Decomposition of Isometries in Nilpotent Lie Groups
Affine Decomposition of Isometries in Nilpotent Lie Groups

Affine Decomposition of Isometries in Nilpotent Lie Groups
Affine Decomposition of Isometries in Nilpotent Lie Groups

... that we just considered has been known since centuries perhaps, if not by the ancient Greeks. There are of course more serious results, which are more like the real motivation for this study. The Euclidean case is just a convenient and understandable example. We shall discuss those more advanced res ...
Non-Associative Local Lie Groups
Non-Associative Local Lie Groups

... global theory. The present paper shows that the local theory is not a simple corollary of the global theory of Lie groups, but has its own set of interesting and delicate geometrical structures. Furthermore, our examples can be viewed as particular instances of nonassociative local group actions on ...
Congruent Triangles
Congruent Triangles

§1: FROM METRIC SPACES TO TOPOLOGICAL SPACES We
§1: FROM METRIC SPACES TO TOPOLOGICAL SPACES We

... Let X and Y be metric spaces, and x ∈ X. A mapping f : X → Y is continuous at x if for every  > 0, there exists δ > 0 such that if d(x, x0 ) < δ, d(f (x), f (x0 )) < . A mapping is said to be continuous if it is continuous at all points x in X. We can recast the definition of continuity in slightl ...
ISOMETRIES BETWEEN OPEN SETS OF CARNOT GROUPS AND
ISOMETRIES BETWEEN OPEN SETS OF CARNOT GROUPS AND

... Theorem 1.1. Let (G, d) be a subFinsler Carnot group. Let Ω1 , Ω2 ⊂ G be two open sets. Let f : Ω1 → Ω2 be an isometry. Then there exists a left translation τ and a group isomorphism φ of G such that f is the restriction to Ω1 of τ ◦ φ, which is an isometry. Note that in the statement above we requi ...
Topology Proceedings - topo.auburn.edu
Topology Proceedings - topo.auburn.edu

... spaces, see Nachbin [9]. Notation and Terminology. Where:S is a pa~tial order on a set X, for x E X: ix denotes the set {y I x y} and lx denotes the set {y I y ~ x}. For P ~ X: iP denotes the set U{ix I x E P} and lP denotes the set U{lx I x E Pl. P is an upper set if P = i P, and is a lower set if ...
Transitive actions of locally compact groups on locally contractible
Transitive actions of locally compact groups on locally contractible

COMPACTIFICATIONS WITH DISCRETE REMAINDERS all
COMPACTIFICATIONS WITH DISCRETE REMAINDERS all

Nonsingular complex instantons on Euclidean spacetime
Nonsingular complex instantons on Euclidean spacetime

... past three decades. It has found applications in a wide variety of research areas, such as differential topology and algebraic geometry [5], representation theory [14], and in the theory of integrable systems [12], to name a few. Usually, one considers the Yang–Mills anti-self-duality (ASD) equations ...
Introduction to symmetric spectra I
Introduction to symmetric spectra I

... Boardman; his construction was written up by Vogt [Vo70]. This category has been extensively studied by algebraic topologists, and as far as I understand, there is no doubt that it is the right category to look at after inverting stable homotopy equivalence. In particular, each of the many different ...
A geometric proof of the Berger Holonomy Theorem
A geometric proof of the Berger Holonomy Theorem

arXiv:1205.2342v1 [math.GR] 10 May 2012 Homogeneous compact
arXiv:1205.2342v1 [math.GR] 10 May 2012 Homogeneous compact

1 2 >

Hermitian symmetric space

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has as an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dual space, a homogeneous space for SU(2) and SL(2,C).Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple compact Lie groups by maximal closed connected subgroups which contain a maximal torus and have center isomorphic to T. There is a complete classification of irreducible spaces, with four classical series, studied by Cartan, and two exceptional cases; the classification can be deduced fromBorel–de Siebenthal theory, which classifies closed connected subgroups containing a maximal torus. Hermitian symmetric spaces appear in the theory of Jordan triple systems, several complex variables, complex geometry, automorphic forms and group representations, in particular permitting the construction of the holomorphic discrete series representations of semisimple Lie groups.
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