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 ...

... 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

... 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 ...

... 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

... 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 ...

... 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

... 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 ...

... 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

... [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 ...

... [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

... 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): ...

... 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

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

... 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

... 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 ...

... 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

... 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 ...

... 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 ...

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 ...

... 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

... 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) ...

... 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

... 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 ...

... 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

... 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 ...

... 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 ...

§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 ...

... 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

... 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 ...

... 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

... 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 ...

... 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 ...

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 ...

... 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

... 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 ...

... 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 ...