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Let f be a contraction mapping from a closed subset F of a Banach space E into F. Then there exists a unique z in F such that f(z)=z.

A family of subsets of a topological space such that every point has a neighborhood that intersects only one of them.

A bounded operator U on a Hilbert space H is called essentially unitary if U^*U-I and UU^*-I are compact operators.

The curvature and torsion functions along a space curve determine it up to an orientation-preserving isometry.

The homeomorphism group of a topological space X is the set of all homeomorphisms f:X->X, which forms a group by composition.

A family of subsets of a topological space such that every point has a neighborhood which intersects only a finite number of them.

A space X is locally pathwise-connected if for every neighborhood around every point in X, there is a smaller, pathwise-connected neighborhood.

An infinite-dimensional differential calculus on the Wiener space, also called stochastic calculus of variations.

A metric space Z^^ in which the closure of a congruence class B(j,m) is the corresponding congruence class {x in Z^^|x=j (mod m)}.

For any two points x,y in X, there is an open set U such that x in U and y not in U or y in U and x not in U. A space fulfilling this axiom is called a T0-space.