equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of closed sets, of closure...

In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems...

various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is...

subspaces of appropriate topological vector spaces by continuous functionals. Banach spaces, introduced by Stefan Banach, are complete normed vector spaces. A...

of infinitely many Woodin cardinals). Quine's system of axiomatic set theory, New Foundations (NF), takes its name from the title ("New Foundations for...

an element of H. As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which...

as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces. The category of Cauchy spaces and Cauchy...

empty set to be open. This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In fact, it...

unique topological space structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and...

From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology...

V is a topological vector space, the quotient space P(V) is a topological space, endowed with the quotient topology of the subspace topology of V \ {0}...

Robert (1964). Axiomatic Analysis. Heath. Krantz, David H.; Luce, R. Duncan; Suppes, Patrick; Tversky, Amos (1971). Foundations of Measurement, Vol...

some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces. The Hahn–Banach theorem is often...

branch of mathematical analysis, the core of which is formed by the study of function spaces, which are some sort of topological vector spaces. Functional...

topology of fundamental groups and covering spaces. A free group G on a set of generators is the fundamental group of a bouquet of circles, a topological graph...

book Company, Incorporated. p. 10. G. Gierz (2006). Bundles of Topological Vector Spaces and Their Duality. Springer. p. 252. ISBN 978-3-540-39437-2....

in January 1998. Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (and...

axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is...

of the first set and elements of the second set are names) and fiber bundles (objects form a topological space, names from another topological space and...

category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion of localization...

addition several other topological properties he defined at the time (he was among the first mathematicians to apply new topological ideas from Hausdorff...

became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable...

space is uncountable. On the other hand, subspaces (of these function spaces) that one typically considers, such as Hilbert spaces (e.g. the space of...

mathematics at the Federal University of Paraná in Curitiba and the title of his 1961 Ph.D. dissertation was Topological spaces and continuous functions. Da Costa's...

Voronoi diagrams and Delaunay triangulations A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments...

areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In...

functional analysis (which studies topological vector spaces that need not have any sense of distance). Formally, a metric space is an ordered pair ( M , d )...

current axiomatic set theory but found it unworkable for undergraduates, so he instead developed the first axioms for the more relevant composition of mappings...

set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids...

algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology...