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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 ...
As in Euclidean space, the fundamental objects in an affine space are called points, which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional ...
Note how in the refinement, no point on the circle is contained in more than two sets, and also how the sets link to one another to form a "chain". Refinement of the cover of a square The top half of the second image shows a cover (colored) of a planar shape (dark), where all of the shape's points are contained in anywhere from one to all four ...
Every space treated in Section "Types of spaces" above, except for "Non-commutative geometry", "Schemes" and "Topoi" subsections, is a set (the "principal base set" of the structure, according to Bourbaki) endowed with some additional structure; elements of the base set are usually called "points" of this space. In contrast, elements of (the ...
A point particle is a 0-brane, of dimension zero; a string, named after vibrating musical strings, is a 1-brane; a membrane, named after vibrating membranes such as drumheads, is a 2-brane. [2] The corresponding object of arbitrary dimension p is called a p -brane, a term coined by M. J. Duff et al. in 1988.
This implies that the Baire space is zero-dimensional with respect to the small inductive dimension (as are all spaces whose base consists of clopen sets.) The above definitions of open and closed sets provide the first two sets Σ 1 0 {\displaystyle \mathbf {\Sigma } _{1}^{0}} and Π 1 0 {\displaystyle \mathbf {\Pi } _{1}^{0}} of the boldface ...
A space in which all components are one-point sets is called totally disconnected. Related to this property, a space X is called totally separated if, for any two distinct elements x and y of X, there exist disjoint open neighborhoods U of x and V of y such that X is the union of U and V. Clearly any totally separated space is totally ...