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The structure theorem is of central importance to TDA; as commented by G. Carlsson, "what makes homology useful as a discriminator between topological spaces is the fact that there is a classification theorem for finitely generated abelian groups".
For every non-empty set S there is a binary operation defined on S that gives it a group structure. [24] (A cancellative binary operation is enough, see group structure and the axiom of choice.) Every free abelian group is projective. [25] Baer's criterion: Every divisible abelian group is injective. [25]
Using the axiom of choice, one can show that for any family S of sets | ⋃S | ≤ | S | × sup { |s| : s ∈ S} (A). [5] Moreover, by Tarski's theorem on choice, another equivalent of the axiom of choice, | X | n = | X | for all finite n (B). Let X be an infinite set and let F denote the set of all finite subsets of X. There is a natural ...
This glossary of biology terms is a list of definitions of fundamental terms and concepts used in biology, the study of life and of living organisms.It is intended as introductory material for novices; for more specific and technical definitions from sub-disciplines and related fields, see Glossary of cell biology, Glossary of genetics, Glossary of evolutionary biology, Glossary of ecology ...
Biological organisation is the organisation of complex biological structures and systems that define life using a reductionistic approach. [1] The traditional hierarchy, as detailed below, extends from atoms to biospheres .
Biological organization spans several scales and are determined based different structures depending on what the system is. [1] Examples of biological systems at the macro scale are populations of organisms. On the organ and tissue scale in mammals and other animals, examples include the circulatory system, the respiratory system, and the ...
Biology portal; Pages in category "Biological theorems" The following 6 pages are in this category, out of 6 total. This list may not reflect recent changes. B. Bet ...
Molecular biology – study of biology and biological functions at the molecular level, with some cross over from biochemistry. Structural biology – a branch of molecular biology, biochemistry, and biophysics concerned with the molecular structure of biological macromolecules. Health sciences and human biology – biology of humans.