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In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. [1] A graphical illustration of a zero-dimensional space is a point. [2]
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, [1] or its generalization to other kinds of mathematical spaces.As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist; conversely ...
The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see the third axiom in the Vector space article). Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F.
Over a commutative ring, a trivial algebra is simultaneously a zero module. The trivial ring is an example of a rng of square zero. A trivial algebra is an example of a zero algebra. The zero-dimensional vector space is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension ...
Also, a three-dimensional projective space is now defined as the space of all one-dimensional subspaces (that is, straight lines through the origin) of a four-dimensional vector space. This shift in foundations requires a new set of axioms, and if these axioms are adopted, the classical axioms of geometry become theorems.
Zero (usually styled as ZERO) was an artist group founded in the late 1950s in Düsseldorf by Heinz Mack and Otto Piene. Piene described it as "a zone of silence and of pure possibilities for a new beginning". [1] In 1961 Günther Uecker joined the initial founders. ZERO became an international movement, with artists from Germany, the ...
This means that an orientation of a zero-dimensional space is a function {{}} {}. It is therefore possible to orient a point in two different ways, positive and negative. Because there is only a single ordered basis ∅ {\displaystyle \emptyset } , a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis.
Such a space is called a Baire space of weight and can be denoted as (). [1] With this definition, the Baire spaces of finite weight would correspond to the Cantor space . The first Baire space of infinite weight is then B ( ℵ 0 ) {\displaystyle B(\aleph _{0})} ; it is homeomorphic to ω ω {\displaystyle \omega ^{\omega }} defined above.