Infinite Matroids

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Solving a long-standing problem of Rado (1966), the first named applicant and coauthors have recently shown that infinite matroids can be axiomatized similarly to finite matroids. Following a series of unsuccessful such attempts until the 1980s, it had become a commonly held belief that this might be impossible. In particular, it was thought that having bases and circuits with their usual properties was incompatible with matroid duality. As a consequence, most authors either disregarded infinite matroids altogether, or imposed a restrictive additional axiom which ensured that bases and circuits existed, but which ruled out duality: the 'finitary axiom' thatAn infinite set is independent as soon as all its finite subsets are independent.The lack of duality resulting from this axiom in effect prevented the development of any theory of infinite matroids along the lines of their finite theory. The newly found axioms for infinite matroids (with duality) will make it possible for the first time to study large classes of known non-finitary matroids, such as the duals of finitary ones. In addition, we hope to find generically infinite applications previously precluded by tlie finitary axiom, such as to Banach or Hilbert spaces.