Zariski topologyIn algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space.
CatThe cat (Felis catus) is a domestic species of small carnivorous mammal. It is the only domesticated species in the family Felidae and is commonly referred to as the domestic cat or house cat to distinguish it from the wild members of the family. Cats are commonly kept as house pets but can also be farm cats or feral cats; the feral cat ranges freely and avoids human contact. Domestic cats are valued by humans for companionship and their ability to kill vermin. About 60 cat breeds are recognized by various cat registries.
Homotopy category of chain complexesIn homological algebra in mathematics, the homotopy category K(A) of chain complexes in an A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the D(A) of A when A is ; unlike the former it is a , and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) turns into isomorphisms any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence.
General topologyIn mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points.
Algebraic varietyAlgebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly.
Associative algebraIn mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field K. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term [[algebra over a field|K-algebra]] to mean an associative algebra over the field K.
AlgebraAlgebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields.
Feral catA feral cat or a stray cat is an unowned domestic cat (Felis catus) that lives outdoors and avoids human contact: it does not allow itself to be handled or touched, and usually remains hidden from humans. Feral cats may breed over dozens of generations and become an aggressive local apex predator in urban, savannah and bushland environments. Some feral cats may become more comfortable with people who regularly feed them, but even with long-term attempts at socialization, they usually remain aloof and are most active after dusk.
Black catA black cat is a domestic cat with black fur that may be a mixed or specific breed, or a common domestic cat of no particular breed. The Cat Fanciers' Association (CFA) recognizes 22 cat breeds that can come with solid black coats. The Bombay breed is exclusively black. All-black fur pigmentation is slightly more prevalent in male cats than female cats. Most black cats have golden irises due to their high melanin pigment content. In popular myths, witches are believed to be associated with black cats.
Persian catThe Persian cat, also known as the Persian longhair, is a long-haired breed of cat characterized by a round face and short muzzle. The first documented ancestors of Persian cats might have been imported into Italy from Khorasan as early as around 1620, however this has not been proven. Instead there is stronger evidence for a longhaired cat breed being exported from Afghanistan and Iran from the 19th century onwards. Widely recognized by cat fancy since the late 19th century, Persian cats were first adopted by the British, and later by American breeders after World War II.
Sphynx catThe Sphynx cat (pronounced , ˈsfɪŋks) also known as the Canadian Sphynx, is a breed of cat known for its lack of fur. Hairlessness in cats is a naturally occurring genetic mutation, and the Sphynx was developed through selective breeding of these animals, starting in the 1960s. According to breed standards, the skin should have the texture of chamois leather, as it has fine hairs, or the cat may be completely hairless. Whiskers may be present, either whole or broken, or may be totally absent.
Mapping cylinderIn mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient where the denotes the disjoint union, and ∼ is the equivalence relation generated by That is, the mapping cylinder is obtained by gluing one end of to via the map . Notice that the "top" of the cylinder is homeomorphic to , while the "bottom" is the space . It is common to write for , and to use the notation or for the mapping cylinder construction.
