# 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.

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

The simple closed curve divides the planeR2 into two bounded (inner) and unbounded (outer) regions with the boundary as the boundary.

Definition

### boundedness of ordered sets

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

If the ordered set (X , ≤) has a maximum element and a minimum element with respect to a partial order ≤, this partial order isbounded order(bounded order) , Or X _ isbounded 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 ) isboundedmeans 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 thebounded 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 numbersR) 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 inR n , consider thatR_ n _ subset _S _ is bounded with respect to this order and this distance It is equivalent to becoming the world.
• Whole real numberRis 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 ofR* _ _ _ _ _ _ _ is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ is a closed set. Generally, the entire bounded subset of the complete distance range becomes compact.

Related item

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Post Date : 2018-01-24 11:30