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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 ...
Alexander horned sphere − A particular embedding of a sphere into 3-dimensional Euclidean space. Antoine's necklace − A topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. Irrational winding of a torus/Irrational cable on a torus; Knot (mathematics) Linear flow on the torus
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.
An open cover of a topological space X is a family of open sets U α such that their union is the whole space, U α = X. The order or ply of an open cover A {\displaystyle {\mathfrak {A}}} = { U α } is the smallest number m (if it exists) for which each point of the space belongs to at most m open sets in the cover: in other words U α 1 ∩ ...
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 ...
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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.