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In mathematics, more specifically algebraic topology, a pair (,) is shorthand for an inclusion of topological spaces:.Sometimes is assumed to be a cofibration.A morphism from (,) to (′, ′) is given by two maps : ′ and : ′ such that ′ =.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological ...
Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. ... ISBN 978-1-4471-2157-2 ...
Albrecht Dold: Lectures on Algebraic Topology, Springer ISBN 3-540-58660-1. Allen Hatcher : Algebraic Topology , Cambridge University Press ISBN 978-0-521-79540-1 . A free electronic version is available on the author's homepage .
An example of a classifying space is that when G is cyclic of order two; then BG is real projective space of infinite dimension, corresponding to the observation that EG can be taken as the contractible space resulting from removing the origin in an infinite-dimensional Hilbert space, with G acting via v going to −v, and allowing for homotopy ...
Shorter examples may fit into the main exposition of the article, such as the discussion at Algebraic number theory § Failure of unique factorization, while others may deserve their own section, as in Chain rule § First example. Multiple related examples may also be given together, as in Adjunction formula § Applications to curves.
A topological algebra over a topological field is a topological vector space together with a bilinear multiplication :, (,)that turns into an algebra over and is continuous in some definite sense.
The topology of = is weak topology: a subset is open iff is open for each k-skeleton . In the language of category theory, the topology on X {\displaystyle X} is the direct limit of the diagram X − 1 ↪ X 0 ↪ X 1 ↪ ⋯ {\displaystyle X_{-1}\hookrightarrow X_{0}\hookrightarrow X_{1}\hookrightarrow \cdots } The name "CW" stands for ...