Math 396. Gluing topologies, the Hausdorff condition, and examples
... Recall that a subset of a topological space is dense if its closure is the entire space. In the setting of metric spaces, this theorem is easily proved by a limiting argument. In general we cannot use limits in X when it is non-Hausdorff (and does not have a countable base of opens around all points ...
... Recall that a subset of a topological space is dense if its closure is the entire space. In the setting of metric spaces, this theorem is easily proved by a limiting argument. In general we cannot use limits in X when it is non-Hausdorff (and does not have a countable base of opens around all points ...
β* - Continuous Maps and Pasting Lemma in Topological Spaces
... not continuous since for the closed set F={b,c} in Y, ( )={b, a} is not closed in X. Theorem 3.5: If a map f : X Y from a topological space X into a topological space Y is g continuous,then it is *-contin uous but not conversely. Proof: Let f : X Y be g-continuous. Let F be any closed set in Y. Then ...
... not continuous since for the closed set F={b,c} in Y, ( )={b, a} is not closed in X. Theorem 3.5: If a map f : X Y from a topological space X into a topological space Y is g continuous,then it is *-contin uous but not conversely. Proof: Let f : X Y be g-continuous. Let F be any closed set in Y. Then ...
K-theory of stratified vector bundles
... Let Y be a V-family. For V ∈ V we denote by Y V the space of all the Vmaps V → Y , endowed with the compact-open topology. A sub-basis of the topology in V Y is given by all the sets of the form NK,U = {f ∈ Y V : f K ⊂ U } with K ⊂ V compact and U ⊂ Y open. For every V-family Y there is a standard p ...
... Let Y be a V-family. For V ∈ V we denote by Y V the space of all the Vmaps V → Y , endowed with the compact-open topology. A sub-basis of the topology in V Y is given by all the sets of the form NK,U = {f ∈ Y V : f K ⊂ U } with K ⊂ V compact and U ⊂ Y open. For every V-family Y there is a standard p ...
Separation axioms
... These are disjoint neighborhoods of x and y respectively showing that (R, Tll ) is T 2. To verify separation axiom T5 , let A, B ⊆ X be two separated sets. Then X − B̄ is an open set and so we can, for each a ∈ A ⊂ X − B̄, find an xa ∈ X such that [a, xa i ⊂ X − B̄ (since the half-open intervals are ...
... These are disjoint neighborhoods of x and y respectively showing that (R, Tll ) is T 2. To verify separation axiom T5 , let A, B ⊆ X be two separated sets. Then X − B̄ is an open set and so we can, for each a ∈ A ⊂ X − B̄, find an xa ∈ X such that [a, xa i ⊂ X − B̄ (since the half-open intervals are ...
Topological Extensions of Linearly Ordered Groups
... A topological space X is called locally compact if for every element x∈ X there exists open neighbourhood U ( x) such that the closure U ( x) is a compact subset of X . Proposition. Let G be a locally compact linearly ordered + with product topology is a topological topological group. Then BG inver ...
... A topological space X is called locally compact if for every element x∈ X there exists open neighbourhood U ( x) such that the closure U ( x) is a compact subset of X . Proposition. Let G be a locally compact linearly ordered + with product topology is a topological topological group. Then BG inver ...
Generalized Continuous Map in Topological Spaces
... Definition 3.1. A map f : X → Y from a topological space (X, τ*) into a topological space (Y, σ*) is called τ*-gc-irresolute if the inverse image of every τ*-g-closed set in Y is τ*-g-closed in X. Theorem 3.2. A map f : X → Y is τ*-gc-irresolute if and only if the inverse image of every τ*-g-open se ...
... Definition 3.1. A map f : X → Y from a topological space (X, τ*) into a topological space (Y, σ*) is called τ*-gc-irresolute if the inverse image of every τ*-g-closed set in Y is τ*-g-closed in X. Theorem 3.2. A map f : X → Y is τ*-gc-irresolute if and only if the inverse image of every τ*-g-open se ...
Continuity in topological spaces and topological invariance
... Therefore Bs (y) ⊆ Br (x) proving that Br (x) is open. Theorem 2 (Equivalence of definitions of Continuity for Metric Spaces). Let (X, dX ), (Y, dY ) be metric spaces, and f : X → Y a function between them. Then f is continuous at a point a ∈ X if and only if for every neighbourhood B of f (a), ther ...
... Therefore Bs (y) ⊆ Br (x) proving that Br (x) is open. Theorem 2 (Equivalence of definitions of Continuity for Metric Spaces). Let (X, dX ), (Y, dY ) be metric spaces, and f : X → Y a function between them. Then f is continuous at a point a ∈ X if and only if for every neighbourhood B of f (a), ther ...
Manifold
In mathematics, a manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded in three dimensional real space, but also the Klein bottle and real projective plane which cannot.Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts). When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds.This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.