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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 ...
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 ...
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 ...
The two structures, "finite-dimensional real or complex linear space" and "finite-dimensional linear topological space", are thus equivalent, that is, mutually underlying. Accordingly, every invertible linear transformation of a finite-dimensional linear topological space is a homeomorphism.
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 ...
The set of geometric primitives is based on the dimension of the region being represented: [1]. Point (0-dimensional), a single location with no height, width, or depth.; Line or curve (1-dimensional), having length but no width, although a linear feature may curve through a higher-dimensional space.
This is a natural way to define a set of points that are relatively close to x. Therefore, a set is a neighborhood of x (informally, it contains all points "close enough" to x) if it contains an open ball of radius r around x for some r > 0. An open set is a set which is a neighborhood of all its points.