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In Euclidean space, such a dilation is a similarity of the space. [2] Dilations change the size but not the shape of an object or figure. Every dilation of a Euclidean space that is not a congruence has a unique fixed point [3] that is called the center of dilation. [4] Some congruences have fixed points and others do not. [5]
A metric space M is compact if every open cover has a finite subcover (the usual topological definition). A metric space M is compact if every sequence has a convergent subsequence. (For general topological spaces this is called sequential compactness and is not equivalent to compactness.) A metric space M is compact if it is complete and ...
The smallest constant is sometimes called the (best) Lipschitz constant [4] of f or the dilation or dilatation [5]: p. 9, Definition 1.4.1 [6] [7] of f. If K = 1 the function is called a short map , and if 0 ≤ K < 1 and f maps a metric space to itself, the function is called a contraction .
If A is an open or closed subset of R n (or even Borel set, see metric space), then A is Lebesgue-measurable. If A is a Lebesgue-measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure. A Lebesgue-measurable set can be "squeezed" between a containing open set and a contained closed set.
Dilation (metric space), a function from a metric space into itself; Dilation (operator theory), a dilation of an operator on a Hilbert space; Dilation (morphology), an operation in mathematical morphology; Scaling (geometry), including: Homogeneous dilation , the scalar multiplication operator on a vector space or affine space; Inhomogeneous ...
In general, when we define metric space the distance function is taken to be a real-valued function. The real numbers form an ordered field which is Archimedean and order complete. These metric spaces have some nice properties like: in a metric space compactness, sequential compactness and countable compactness are equivalent etc.
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.
A metric space is called complete if all Cauchy sequences converge. Every incomplete space is isometrically embedded, as a dense subset, into a complete space (the completion). Every compact metric space is complete; the real line is non-compact but complete; the open interval (0,1) is incomplete. Every Euclidean space is also a complete metric ...