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A degree two map of a sphere onto itself. In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping.
This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it. Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country.
In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can often be used to prove existence of a second ...
In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres (,) (,) or equivalently to boundary-sphere-preserving continuous maps between balls (,) (,) to boundary-sphere-preserving maps between balls in a Banach space: ((), ()) ((), ()), assuming that the map is of the form = where is the identity map and is some compact ...
Pavel Urysohn. In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma [1]) states that any real-valued, continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
A continuous linear action ∗ : G × H → H, gives rise to a continuous map ρ ∗ : G → H H (functions from H to H with the strong topology) defined by: ρ ∗ (g)(v) = ∗(g,v). This map is clearly a homomorphism from G into GL(H), the bounded linear operators on H. Conversely, given such a map, we can uniquely recover the action in the ...
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Every continuous map from a compact space to a Hausdorff space is both proper and closed. Every surjective proper map is a compact covering map. A map f : X → Y {\displaystyle f:X\to Y} is called a compact covering if for every compact subset K ⊆ Y {\displaystyle K\subseteq Y} there exists some compact subset C ⊆ X {\displaystyle C ...