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Slides
Slides

... The class of logically valid ∀∃p sentences in the language of metric space is decidable. Proof: Taking the negation we look for a decision procedure for the satisfiability of ∃∀p sentences. Let φ ≡ ∃x1 , . . . , xn . ∀ȳ /Qz̄. ψ. Thanks to previous theorem, φ is satisfiable iff exists an interpretat ...
Math 396. Paracompactness and local compactness 1. Motivation
Math 396. Paracompactness and local compactness 1. Motivation

... often adequate for many purposes. It is this feature of Rn that we seek to generalize. 2. Definitions Let X be a topological space. Definition 2.1. The space X is locally compact if each x ∈ X admits a compact neighborhood N . If X is locally compact and Hausdorff, then all compact sets in X are clo ...
Compact operators on Banach spaces
Compact operators on Banach spaces

... y1 , . . . , yn ∈ X linearly independent modulo (T − λ)X, by Hahn-Banach there are η1 , . . . , ηn ∈ X ∗ vanishing on the image (T − λ)X and ηi (yj ) = δij . Such ηi are in the kernel of the adjoint (T − λ)∗ . We know T ∗ is compact, so ker(T − λ)∗ is finite-dimensional. We’ve proven that injectivit ...
Group actions in symplectic geometry
Group actions in symplectic geometry

2.5 Proving Statements About Segments
2.5 Proving Statements About Segments

... 1. Reflexive Property: AB  AB A  A 2. Symmetric Prop: If AB  CD, then CD  AB If A  B, then B  A 3. Transitive Prop: If AB  CD and CD  EF, then AB  EF IF A  B and B  C, then A  ...
(pdf)
(pdf)

DISCONTINUOUS GROUPS AND CLIFFORD
DISCONTINUOUS GROUPS AND CLIFFORD

... discontinuously on a homogeneous manifold G/H if H is noncompact. We discuss recent developments in the theory of discontinuous groups acting on a homogeneous manifold G/H where G is a real reductive Lie group and H is a noncompact reductive subgroup. Geometric ideas of various methods together with ...
Proper actions on topological groups: Applications to quotient spaces
Proper actions on topological groups: Applications to quotient spaces

... addition, X is a proper G-space then ρe is, in fact, a compatible metric on X/G [23, Theorem 4.3.4]. For a closed subgroup H ⊂ G, by G/H we will denote the G-space of cosets {gH| g ∈ G} under the action induced by left translations. A locally compact group G is called almost connected if the quotien ...
Math 525 More notes about compactness (sec 26
Math 525 More notes about compactness (sec 26

... is not second countable, since L contains uncountably many disjoint open sets (choose an open interval in each column). Thus (R, τu ) is homeomorphic to a subspace of L (indeed, to the first column), but L is not homeomorphic to a subspace of R, since second countability is hereditary. Also, L is no ...
ON MINIMAL, STRONGLY PROXIMAL ACTIONS OF LOCALLY
ON MINIMAL, STRONGLY PROXIMAL ACTIONS OF LOCALLY

... exists a unique maximal irreducible affine G-representation V (G) which is universal in the sense that for any irreducible affine G-representation W there exists a unique continuous Gequivariant affine surjection qW : V (G) → W . It follows that V (G) = P(B(G)). The universal boundary B(G) is a Haus ...
26 - HKU
26 - HKU

On Top Spaces
On Top Spaces

... for constructing top generalized subgroups is considered. Connected component of an identity as a top generalized normal subgroup is studied. A criterion for the connectedness of an inverse image of an identity is deduced. A condition for the separability of a top space is presented. Top generalized ...
spaces in which compact sets have countable local bases
spaces in which compact sets have countable local bases

... Let X = f(x, y) £ R2: y > OÍ. Define U by (Ma, b) = |(a, y) £ X: \y- b\< l/n\ ¡7n(a, 0) = |(x, y) £ X: y < l/n, U defines ...
Section 29. Local Compactness - Faculty
Section 29. Local Compactness - Faculty

... is related to the fact that the diameter of the Riemann sphere is 2. This metric induces the topology on C ∪ {∞} described in Theorem 29.1 For details about the metric and some related projections, see my Complex Analysis 1 (MATH 5510) notes: http://faculty.etsu.edu/gardnerr/5510/notes/I-6.pdf. ...
File - Miss Pereira
File - Miss Pereira

... equal lengths are congruent and that angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. ...
Part I Linear Spaces
Part I Linear Spaces

... X is compact if for any open covering there is a finite subcover. A space is locally compact if every point has a compact neighborhood. Properties: 1. (finite intersection property) For any family of closed set with nonempty intersection we could find a finite subfamily with nonempty intersection. I ...
PROPERTIES OF FINITE-DIMENSIONAL GROUPS Topological
PROPERTIES OF FINITE-DIMENSIONAL GROUPS Topological

... manifold M, does this imply that G is a manifold, and hence a Lie group? In connection with 2 and 2', Zippin and the author have shown [10; 8; 16] that any compact connected group acting effectively on a three-dimensional manifold M must be a Lie group. If M = Ez, we showed further that G must be eq ...
New examples of totally disconnected locally compact groups
New examples of totally disconnected locally compact groups

... Y. Shalom and G. Willis, Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity, arXiv:0911.1966 G. Willis, The structure of totally disconnected, locally compact groups, Mathematische ...
Cantor`s Theorem and Locally Compact Spaces
Cantor`s Theorem and Locally Compact Spaces

... One of the easiest cases in which you can determine all the compact subsets of a topological space is the Euclidean spaces Rn . Definition A subset Y of a metric space X is bounded if there exists C > 0 such that d(x, y) < C for all x, y ∈ Y . Theorem 3. (Heine-Borel Theorem) Let Y ⊂ Rn . Then Y is ...
On the average distance property of compact connected metric spaces
On the average distance property of compact connected metric spaces

... Consider the n-sphere, S n, with metric d inherited from the Euclidean metric on R n+l, and with /t0 the normalisation of the usual n-dimensional measure on S n. Then for each y and e in S n there exists an isometry T as in the above lemma, namely, a suitable rotation. Hence we obtain Theorem 2. Let ...
1 Preliminaries
1 Preliminaries

... A topological space is irreducible if any (and hence all) of the following conditions hold: • The space is not the union of two proper non-empty closed sets • Any two non-empty open sets intersect • Any non-empty open set is dense A subset of a topological space irreducible if it is irreducible with ...
Exercise Sheet 4 - D-MATH
Exercise Sheet 4 - D-MATH

... Tom Ilmanen ...
Universal cover of a Lie group. Last time Andrew Marshall
Universal cover of a Lie group. Last time Andrew Marshall

... Consider the space Gr(M ) = P (M )/ ∼ consisting of all smooth paths γ : [0, 1] → M modulo the equivalence relation of endpoint fixing homotopies. We can compose some elements of the groupoid, but, unlike a group, not all elements can be composed. Specifically, we can compose two elements if and onl ...
Locally Compact Hausdorff Spaces
Locally Compact Hausdorff Spaces

... Let X be a topological space and F ⊂ C (X ). F is equicontinuous at x ∈ X if for every  > 0 there is an open set U containing x such that |f (x) − f (y )| <  for all y ∈ U and f ∈ F. F is equicontinuous if it is equicontinuous at each point x ∈ X . F is pointwise bounded if {f (x) : f ∈ F} is a b ...
Compactness (1) Let f : X → Y be continuous and X compact. Prove
Compactness (1) Let f : X → Y be continuous and X compact. Prove

... ...
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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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