The following topics are immediate subtopics of Field Of Topology.

Term: Affinely Independent
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 Definition: A set (x_0, ... , x_k} of points of R^N is said to be geometrically independent, or affinely independent, if the equations sum[(a_i)(x_i)]=0 (with i=0 to i=k), and sum(a_i)=0 (with i=0 to i=k) hold...

Term: Ascoli's Theorem
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 Definition: Let X be a space and let (Y, d) be a metric space. Give C(X, Y) the topology of compact convergence; let F be a subset of C(X, Y). a) If F is equicontinuous under d and the set F_d = {f(a)f in F...

Term: Baire Category Theorem
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 Definition: If X is a compact Hausdorff space or a complete metric space, then X is a Baire space.

Term: Baire Space
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 Definition: A space X is said to be a Baire space if the following condition holds: Given any countable collection {A_n} of closed sets of X each of which has empty interior in X, their union A_n also has empt...

Term: Basis
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 Definition: If X is a set, a basis for a topology on X is a collection B of subsets of X (called basis elements) such that: (1) For each x in X, there is at least one basis element b containing x. (2) If x bel...

Term: Bounded (Set)
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 Definition: Let X be a metric space with metric d. A subset A of X is said to be bounded if there is some number M such that d(x, y)<=M for every pair x, y of points of A.

Term: Box Topology
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 Definition: Let {X_a} be a family of indexed topological spaces. Let Y denote the generalized Cartesian product of the sets X_a. Let B denote the set of all products of open sets of the corresponding spaces....

Term: Cauchy Integral Formula
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 Definition: Let C be a simple closed piecewisedifferentiable curve in the complex plane. Let B be the bounded component of R^2  C. If F(z) is analytic in an open set omega that contains B and C, then for e...

Term: Cauchy Sequence
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 Definition: Let (X, d) be a metric space. A sequence (x_n) of points of X is said to be a Cauchy sequence in (X, d) if it has the property that given epsilon greater than 0, there is an integer N such that d(...

Term: Closed Map
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 Definition: A map f: X > Y is said to be a closed map if for each closed set A of X. the set f(A) is closed in Y.