On feebly compact shift-continuous topologies on the semilattice
... Next we shall show that the family VSis discrete in X. Indeed, since the family V is locally finite S in X, by Theorem 1.1.11 of [8] the union V is a closed subset of X, and hence any point x ∈ X \ V has an open neighbourhood which does not intersect the elements of the family V . If x ∈ clX (Vi ) f ...
... Next we shall show that the family VSis discrete in X. Indeed, since the family V is locally finite S in X, by Theorem 1.1.11 of [8] the union V is a closed subset of X, and hence any point x ∈ X \ V has an open neighbourhood which does not intersect the elements of the family V . If x ∈ clX (Vi ) f ...
Lindelo¨f spaces C(X) over topological groups - E
... answered this question positively but under PFA. Eberlein compact spaces provide a large class of spaces with countable tightness, and it is known that homogeneous Eberlein compact spaces are first countable [4, III.3.10], but non-metrizable homogeneous Eberlein compact spaces do exist, see [46]. In ...
... answered this question positively but under PFA. Eberlein compact spaces provide a large class of spaces with countable tightness, and it is known that homogeneous Eberlein compact spaces are first countable [4, III.3.10], but non-metrizable homogeneous Eberlein compact spaces do exist, see [46]. In ...
COMPACTIFICATIONS AND REMAINDERS OF MONOTONICALLY
... showed in [14] that weak orthocompactness is preserved in various topological operations. We can use his results and Theorem 1.1 to establish invariance properties for the class LM of locally compact spaces with a monotonically normal compactification. This way we see, for instance, that the class i ...
... showed in [14] that weak orthocompactness is preserved in various topological operations. We can use his results and Theorem 1.1 to establish invariance properties for the class LM of locally compact spaces with a monotonically normal compactification. This way we see, for instance, that the class i ...
2 - Ohio State Department of Mathematics
... Word hyperbolicity We will show below that by using the strict hyperbolization technique of Charney–Davis [4], one can arrange for nontriangulable aspherical manifolds of dimension ≥ 6 to have word hyperbolic fundamental groups. So, in this paragraph h(K) is the strict hyperbolization functor of [4] ...
... Word hyperbolicity We will show below that by using the strict hyperbolization technique of Charney–Davis [4], one can arrange for nontriangulable aspherical manifolds of dimension ≥ 6 to have word hyperbolic fundamental groups. So, in this paragraph h(K) is the strict hyperbolization functor of [4] ...
On topological groups via a-local functions - RiuNet
... where τ (x) = {U ∈ τ | x ∈ U }. A Kuratowski closure operator Cl∗ (.) for a topology τ ∗ (τ, I), called the ∗-topology, which is finer than τ is defined by Cl∗ (A) = A ∪ A∗ (τ, I), when there is no chance of confusion. A∗ (I) is denoted by A∗ and τ ∗ for τ ∗ (I, τ ). X ∗ is often a proper subset of ...
... where τ (x) = {U ∈ τ | x ∈ U }. A Kuratowski closure operator Cl∗ (.) for a topology τ ∗ (τ, I), called the ∗-topology, which is finer than τ is defined by Cl∗ (A) = A ∪ A∗ (τ, I), when there is no chance of confusion. A∗ (I) is denoted by A∗ and τ ∗ for τ ∗ (I, τ ). X ∗ is often a proper subset of ...
Introduction to Topology
... continuous at a ∈ X if and only if for each ² > 0 there exists δ > 0 such that f (Bδ (a)) ⊆ B² (f (a)) (or equivalently, Bδ (a) ⊆ f −1 [B² (f (a))]). 4.3 Theorem. Let (X, d) be a metric space and let (an ) be a sequence in X. Then limn an = a if and only if for each ² > 0 there exists a positive int ...
... continuous at a ∈ X if and only if for each ² > 0 there exists δ > 0 such that f (Bδ (a)) ⊆ B² (f (a)) (or equivalently, Bδ (a) ⊆ f −1 [B² (f (a))]). 4.3 Theorem. Let (X, d) be a metric space and let (an ) be a sequence in X. Then limn an = a if and only if for each ² > 0 there exists a positive int ...
The Euler characteristic of an even
... merging two graphs along a simple vertex. It also assures that if two geometric 2-dimensional graphs are glued along a contractible circle and the discs bounded by the circles are taken out on both sides, then the genus g = 1 − χ(G)/2 is additive as long as we apply it to twodimensional geometric gr ...
... merging two graphs along a simple vertex. It also assures that if two geometric 2-dimensional graphs are glued along a contractible circle and the discs bounded by the circles are taken out on both sides, then the genus g = 1 − χ(G)/2 is additive as long as we apply it to twodimensional geometric gr ...
Francesca Ferrari
... To each Niemeier lattice of type X it is possible to associate an Umbral group G X , de ned as the quotient of the automorphism group of the lattice by the Weyl group of X . ...
... To each Niemeier lattice of type X it is possible to associate an Umbral group G X , de ned as the quotient of the automorphism group of the lattice by the Weyl group of X . ...
Regular Hypersurfaces, Intrinsic Perimeter and Implicit Function
... If p, q ∈ G, their cc-distance dc (p, q) is dc (p, q) = inf {T > 0 : γ : [0, T ] → G is subunit, γ(0) = p, γ(T ) = q} . (2) The fact, that under assumption (1), dc (p, q) is finite for any p, q is the content of Chow theorem (see e.g. [6] or [28]). We recall that the topology induced on Rn by dc is t ...
... If p, q ∈ G, their cc-distance dc (p, q) is dc (p, q) = inf {T > 0 : γ : [0, T ] → G is subunit, γ(0) = p, γ(T ) = q} . (2) The fact, that under assumption (1), dc (p, q) is finite for any p, q is the content of Chow theorem (see e.g. [6] or [28]). We recall that the topology induced on Rn by dc is t ...
the isoperimetric problem on some singular surfaces
... Hugh Howards, Michael Hutchings, and Frank Morgan [9] provide a survey of leastperimeter enclosures. Some higher dimensional ambients with conical singularities are treated in [15] and [3]. 2. Existence and regularity We consider piecewise smooth (stratified) n-dimensional closed submanifolds M of R ...
... Hugh Howards, Michael Hutchings, and Frank Morgan [9] provide a survey of leastperimeter enclosures. Some higher dimensional ambients with conical singularities are treated in [15] and [3]. 2. Existence and regularity We consider piecewise smooth (stratified) n-dimensional closed submanifolds M of R ...
TOPOLOGICAL GROUPS The purpose of these notes
... Since p is an open map by (2), and Tg and ι are continuous, this means Tp(g) and ι0 must also be continuous, making G/H a topological group. (7): By Proposition 1.4, H = {1} is a subgroup of G. H is then the minimal closed subgroup of G containing 1, while for any x ∈ G, xHx−1 is also a closed subgr ...
... Since p is an open map by (2), and Tg and ι are continuous, this means Tp(g) and ι0 must also be continuous, making G/H a topological group. (7): By Proposition 1.4, H = {1} is a subgroup of G. H is then the minimal closed subgroup of G containing 1, while for any x ∈ G, xHx−1 is also a closed subgr ...
The inverse map of a continuous bijective map might not be
... Fact: Let X�and Y be two topological spaces. Assume X is compact and assume that Y is Hausdorff. Let f : X → Y be a continuous map such that f is also bijective. Then f −1 is a continuous map from Y to X. If X�is not assumed to be compact, then for a bijective map f : X → Y , f being continuous canno ...
... Fact: Let X�and Y be two topological spaces. Assume X is compact and assume that Y is Hausdorff. Let f : X → Y be a continuous map such that f is also bijective. Then f −1 is a continuous map from Y to X. If X�is not assumed to be compact, then for a bijective map f : X → Y , f being continuous canno ...
