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The concept of a distance, or metric, between the elements of a set is well defined in topology. Let B be any set with x, y, z Є B, and let D be a function from the Cartesian product, B X B, into the set of real numbers, R. D is called a metric on B if the following four statements hold:
1) D(x,y) ≥ 0 for all x, y
2) D(x,y) = D(y,x) for all x, y
3) D(x,y) = 0 if and only if x = yfor all x, y
4) D(x,y) + D(y,z) ≥ D(x,z) for all x, y, z
In this case, the set B having metric D is called a metric space, and is often notated as B, D.
For more information and related definitions, see the links below.
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Define sets in maths?
A set is a collection of well-defined and distinct objects.
What is the difference between set and space in mathematics?
A space is a set with structure. A number of different kinds of mathematical structures (or topologies) exist, including metrics, norms, and inner products. Sets paired with each of these result in a different kinds of spaces, each with a host of interesting properties. Examples of metric spaces include 2-dimensional Euclidean space (as in the surface of a flat sheet of paper), 3-dimensional Euclidean space (a simplification of the world we live in), the Minkowski space (our 3-D world subjected to Einstein's special relativity), and elliptic geometry (which can be used to measure the distances between locations on the surface of the Earth). There also exist topological spaces that are not metric spaces, i.e. spaces that do not have a strict notion of distance between their points. The same set may be paired with different topologies (or different metrics, if applicable), and each of these pairings should be thought of as forming distinct topological (or metric) spaces. There are no spaces that are not sets. On the other hand, any set not paired with a topology is not a space. One can, however, pair any set with the trivial metric d(x,y) = { 0 iff x = y, 1 otherwise } to arrive at a trivial topology. Disregarding this, it is easy to imagine sets that are not spaces, such as for example the set of all automobile models with model year 2013.
How do you define sets in math?
A set is a collection of well defined and distinct objects, considered as an object in its own right.
Is any second category space a Baire space?
Yes, any second category space is a Baire space. A topological space is considered to be of second category if it cannot be expressed as a countable union of nowhere dense sets. Baire spaces are defined by the property that the intersection of countably many dense open sets is dense. Therefore, since second category spaces avoid being decomposed into countable unions of nowhere dense sets, they satisfy the conditions to be classified as Baire spaces.
How many data sets does a outliers have?
Each outlier is a single point in the outcome space.
