The following topics are immediate subtopics of Algebraic Topology.

Term: Antipode
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 Definition: If x is a point of S^n (the unit nsphere), then its antipode is the point x.

Term: Antipode Preserving
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 Definition: A map h: S^n > S^m is said to be antipodepreserving if h(x) = h(x) for all x in S^n.

Term: Arc
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 Definition: An arc A is a space homeomorphic to the unit interval [0, 1]. The end points of A are the two points and q of A such that Ap and Aq are connected; the other points of A are called interior poin...

Term: Base Point
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 Definition: In a loop, the initial point of the path is the same as the end point of the path; it is called the base point.

Term: Betti Number
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 Definition: If G is abelian and finitely generated, then there is a fundamental theorem to the effect that G is the direct sum of two subgroups, H and T. H is a free abelian of finite rank, and T is a subgroup...

Term: Borsuk Lemma
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 Definition: Let a and b be points of S^2. Let A be a compact space, and let f: A > S^2 {a, b} be a continuous map. If f is nulhomotopic, then a and b lie in the same component of S^2  f(A).

Term: Borsuk Ulam Theorem
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 Definition: Given a continuous map f: S^(n+1) > R^(n+1), there is a point x of S^(n+1) such that f(x) = f(x).

Term: Brouwer Fixed Point Theorem
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 Definition: Any continuous function g: B^n > B^n has a fixed point, where B^n is the unit nball.

Term: Coherent
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 Definition: Let X be a space that is the union of the subspaces X_a, for a in J. The topology of X is said to be coherent with the subspaces X_a provided a subset C of X is closed in X if C intersect X_a is c...

Term: Complete Set Of Relations
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 Definition: Given G, suppose we are given a family of generators {x_a}_(a in J) of generators for G. Let F be the the free group on the elements {x_a}. Then the obvious map h(x_a) = x_a of these elements into ...