Existence world

Existence world meeting, a bottom expressed me on world meeting schematically the top. But the lower one shall meander across a frame to the right side.

When it has finiteness about the certain "size of the diameter", a meeting is existence world (worth to say British: bounded) or it is said that it is existence world meeting (worth meeting to say, bounded set) in mathematics. It is said that the meeting that is not existence world is me on world (metaphor detour, unbounded).

The simple closed curve divides plane R² into the domain of two that are existence world (the inside) and me on world (the outside) in it as a border.

Table of contents

Definition

Existence world characteristics of the ordered set

I regard subset A which is not empty as ordered set (X,≤). When former L of X satisfies a≤L about any former a of A, L is called the upper world of form (upper bound) of A, and it is said that A having the upper world of form is which it is existence world or "is controlled on the top by the top" (bounded [from] above). It is the cause of X again l But, about any former a of A l l is called this world (lower bound) of A if I satisfy ≤a, and it is said with (bounded [from] below) which A having the earth is existence world below or "is controlled by the bottom".

It is said that the meeting held in check by top and bottom both sides is existence world.

If ordered set (X,≤) has most Omoto and the smallest cause about half order ≤, this half order is existence world order (bounded order) or it is said that X is existence world ordered set (bounded poset). For ordered set X with existence world order, (S,≤) which limited order in subset S does not necessarily become the existence world order.

Existence world characteristics of the metric space

When subset S of the metric space (M, d) is existence world, I say that S can cover with a ball with a limited radius. In other words, it is former x and positive number r of M > For former s of S where is arbitrary with 0 d (x, s) When a thing becoming <r exists, it is said that S is existence world.

When M is existence world with expectation of a subset of M in itself, d is called existence world distance function (bounded metric) and calls M existence world metric space (bounded metric space).

An example and property

• Open interval (a, b) and closed interval [a, b] consisting of real numbers are existence world as an ordered set (about the normal Euclid distance) (about the big things and small things relations of normal real number) as a metric space.
• If the meeting (subset of meeting R which the whole real number makes) consisting of real numbers is existence world, existence world section including it exists.
• Generally, subset S of the Rⁿ is equivalent with it being about existence world and a thing and this distance that it is about this order in existence world when I put big things and small things-related prompt shipment order and normal Euclid distance in Rⁿ and think.
• Whole real number R is not existence world (Archimedes characteristics).
• The existence world set of R that is not empty has the upper limit (least upper bound) and the lower limit (infimum).
• The existence world set of the Euclidean space Rⁿ is all existence world. If it is a closed set, the existence world set of the Rⁿ in particular is compact. Generally, all existence world subset of the perfection metric space becomes compact.

Allied item

• All existence world (previous compact)
• Ordered set
• Directed meeting

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This article is taken from the Japanese Wikipedia Existence world

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