# Bound

The top is a bounded set, and the bottom is a schematic representation of an unbounded set. However, the lower one shall extend to the right side beyond the frame.

Is**bounded**(Yu-kai,: *bounded*), or**bounded set**(Yu-kai, _bounded_set ) Refers to when it has finiteness with respect to certain "magnitude of deliveries". Unbound sets are**unbounded** It is said to be unbounded.

The simple closed curve divides the plane**R**2 into two bounded (inner) and unbounded (outer) regions with the boundary as the boundary.

## table of contents

Definition

### boundedness of ordered sets

(* X *, ≤) and its non-empty subset _ A *. For _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * _ * * ___ _ * _ * _ _ _ upper bound**(upper bound), _A _ with upper bound bounded bound**on bound** Or "bounded [from] above". Also, the _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ If _l _ ≤ _ a _ is satisfied, _ l _ is called**lower bound

**(lower bound) of _ A _, and _ A _ having a lower bound**Bounded below**or "bounded [from] below".

The set which can be suppressed from both the upper and lower sides is said to be**bounded**.

If the ordered set (*X *, ≤) has a maximum element and a minimum element with respect to a partial order ≤, this partial order is**bounded order**(bounded order) , Or *X _ is bounded ordered set(bounded poset). For ordered set _X _ with bounded order, subset _S * (

*S*, ≤) which restricted the order to the bounded order is not necessarily in the bounded order.

### boundedness of the distance space

A subset *S _ of (_M *, _ d *) is boundedmeans that _ S _ can be covered by a sphere having a finite radius. That is, When _x _ and positive number _ r *> 0 exist and there is something such that _ d _ (

*x*, _ s

*) <*r _ for an arbitrary _ S _ element _ s

*It is said that _S*is bounded.

When *M _ is itself bounded as a subset of _ M *, _ d _ is called the**bounded distance function**(bounded metric) and _ M _ **Called bounded distance space**(bounded metric space).

Examples and properties

- If a set consisting of real numbers (a set of all real numbers
**R**) is bounded, then there is a bounded interval containing it. - In general, when considering the product product order of magnitude relation and ordinary Euclidean distance in
**R***n*, consider that**R**_ n _ subset _S _ is bounded with respect to this order and this distance It is equivalent to becoming the world. - Whole real number
**R**is not bounded ().*The unbounded bounded set of R**has (minimum upper bound) and (maximum lower bound). The bounded set of*R****_ n _ is entirely bounded. In particular, the bounded set of**R*_*_*_*_*_*_*_ is _*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*is a closed set. Generally, the entire bounded subset of the complete distance range becomes compact.

Related item

- (Previous Compact)
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Post Date : 2018-01-24 11:30

This article is taken from the Japanese Wikipedia **Bound**

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