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Sometimes, if each function in a normal family F satisfies a particular property (e.g. is holomorphic), then the property also holds for each limit point of the set F. More formally, let X and Y be topological spaces. The set of continuous functions : has a natural topology called the compact-open topology. A normal family is a pre-compact ...
Let X and Y be two metric spaces, and F a family of functions from X to Y.We shall denote by d the respective metrics of these spaces.. The family F is equicontinuous at a point x 0 ∈ X if for every ε > 0, there exists a δ > 0 such that d(ƒ(x 0), ƒ(x)) < ε for all ƒ ∈ F and all x such that d(x 0, x) < δ.
Other families of functions obtained by picking one arbitrary function from each T(X) → T(X) would not have such uniformity. They are called " ad hoc polymorphic functions" . Parametricity is the abstract property enjoyed by the uniformly acting families such as twice , which distinguishes them from ad hoc families.
The compact-open topology on the space of continuous functions from to has for a subbase the set of functions (,) = {: ()} where is compact and is an open subset of . Suppose that ( X , τ ) {\displaystyle (X,\tau )} is a Hausdorff topological space with X {\displaystyle X} containing two or more elements (for example, X = R {\displaystyle X ...
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel , and give conditions under which a family of holomorphic functions is normal .
Cerf showed that a one-parameter family of functions between two Morse functions can be approximated by one that is Morse at all but finitely many degenerate times. The degeneracies involve a birth/death transition of critical points, as in the above example when, at t = 0 {\displaystyle t=0} , an index 0 and index 1 critical point are created ...
“The thing that has impressed me with this team is that they don’t get down,” Chiefs head coach Andy Reid said Monday. “That’s not what they are.
The ErbB protein family or epidermal growth factor receptor (EGFR) family is a family of four structurally related receptor tyrosine kinases. Insufficient ErbB signaling in humans is associated with the development of neurodegenerative diseases , such as multiple sclerosis and Alzheimer's disease . [ 12 ]