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Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include the Cantor space and Baire space. Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers where = {,} is given the discrete topology.
For an n-dimensional lattice, identifying n linearly independent commas reduces the dimension of the lattice to zero, meaning that the number of pitches in the lattice is finite; mathematically, its quotient is a finite abelian group. This zero-dimensional set of pitches is a periodicity block.
The simplest pitch space model is the real line. A fundamental frequency f is mapped to a real number p according to the equation = + (/) This creates a linear space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and middle C is assigned the number 60, as it is in MIDI. 440 Hz is the standard frequency of 'concert A', which ...
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.
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 ...
Examples of musical lattices include the Tonnetz of Euler (1739) and Hugo Riemann and the tuning systems of composer-theorists Ben Johnston and James Tenney. Musical intervals in just intonation are related to those in equal tuning by Adriaan Fokker's Fokker periodicity blocks. Many multi-dimensional higher-limit tunings have been mapped by Erv ...
It is zero-dimensional and totally disconnected. It is not locally compact. It is universal for Polish spaces in the sense that it can be mapped continuously onto any non-empty Polish space. Moreover, any Polish space has a dense G δ subspace homeomorphic to a G δ subspace of the Baire space.
The ring R = k[x,y]/(x 2, y 2, xy) is a 0-dimensional Cohen–Macaulay ring that is not a Gorenstein ring. In more detail: a basis for R as a k-vector space is given by: {,,}. R is not Gorenstein because the socle has dimension 2 (not 1) as a k-vector space, spanned by x and y.