Interval (math)

In analysis , order topology and related areas of mathematics, an interval is a “connected” subset of a total (or linear) ordered set of carriers (for example the set of real numbers ). A (limited) interval consists of all elements that can be compared in terms of size with two limiting elements of the carrier set, the lower limit and the upper limit of the interval, and which in the sense of this comparison are between ${\ displaystyle \ mathbb {R}}$${\ displaystyle x}$ ${\ displaystyle a}$${\ displaystyle b}$the limits. The limits of the interval may belong to the interval (closed interval, )${\ displaystyle a \ leq x \ leq b}$ do not belong (open interval )${\ displaystyle a or partially belong (semi-open interval ; ).${\ displaystyle a \ leq x ${\ displaystyle a

Connected here means: If two objects are contained in the subset, then all objects in between (in the carrier set) are also contained in it. The most important examples of carrier sets are the sets of real, rational, whole, and natural numbers. In the cases mentioned, and more generally whenever a difference between two elements of the set of carriers is explained, the difference between the upper and lower limit of the interval ( ) is called the length of the interval or, for short , the length of the interval; The term interval diameter is also used for this difference . If an arithmetic mean of the interval limits is declared, this is called the interval center . ${\ displaystyle ba}$

Examples

In the set of natural numbers
${\ displaystyle \ {5,6,7,8,9 \}}$

In this case of a discrete set, the elements of the interval are contiguous.

In the set of real numbers
${\ displaystyle [0,1] = \ {x \ in \ mathbb {R} \ mid 0 \ leq x \ leq 1 \}}$,

the set of all numbers between 0 and 1, including the endpoints 0 and 1.

Trivial examples of intervals are the empty set and sets that have exactly one element. If one does not want to include these, then one speaks of real intervals.

The set can also be viewed as a subset of the carrier set of the real numbers. In this case, it is not an interval because the set does not include, for example, the unnatural numbers between 6 and 7. ${\ displaystyle \ {5,6,7,8,9 \}}$

The carrier set of real numbers so far plays a special role among the mentioned support levels for intervals when they order complete (see p. A. Dedekind cut ). Intervals in this case are exactly within the meaning of topology related subsets.

Designations and spellings

An interval can be limited (on both sides) or - even on one side - unlimited . It is clearly determined by its lower and upper interval limit if it is also specified whether these limits are included in the interval.

There are two different commonly used interval notations:

• The more common of the two uses square brackets for limits that belong to the interval and round brackets for limits that do not belong to the interval. The square brackets correspond to a weak inequality sign ≤. The round brackets () correspond to a strong inequality sign <.
• The other spelling uses outward-facing (mirrored) square brackets instead of round brackets. In the following, both notations are shown and compared to the notation of quantities :

Limited intervals

Be . A restricted interval with the lower limit and the upper limit is completed , when it contains both of the limits, and open , when both boundaries are not included. A restricted interval is called half-open if it contains exactly one of the two interval limits . ${\ displaystyle a ${\ displaystyle a}$${\ displaystyle b}$

Closed interval

${\ displaystyle [a, b]: = \ {x \ in \ mathbb {R} \ mid a \ leq x \ leq b \}}$

The interval contains both and . ${\ displaystyle a}$${\ displaystyle b}$

An interval is compact if and only if it is closed and bounded.

Open interval

${\ displaystyle (a, b) = {] a, b [}: = \ {x \ in \ mathbb {R} \ mid a

The interval contains neither nor . The notation is the traditional one, while it goes back to Bourbaki . ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle (a, b)}$${\ displaystyle {] a, b [}}$

Half-open (more precisely right-open) interval

${\ displaystyle [a, b) = {[a, b [}: = \ {x \ in \ mathbb {R} \ mid a \ leq x

The interval contains but does not . ${\ displaystyle a}$${\ displaystyle b}$

Half-open (more precisely left-open) interval

${\ displaystyle (a, b] = {] a, b]}: = \ {x \ in \ mathbb {R} \ mid a

The interval does not contain , but it does . ${\ displaystyle a}$${\ displaystyle b}$

In the case of and this is called the open unit interval and the closed unit interval . ${\ displaystyle a = 0}$${\ displaystyle b = 1}$${\ displaystyle (a, b)}$${\ displaystyle [a, b]}$

Unlimited intervals

If the interval limit is missing on one side, i.e. there should be no limit there, one speaks of an (on this side) unrestricted interval. Usually the known symbols and “replacement” interval limits are used for this, which themselves never belong to the interval (hence the spelling with round brackets). In some literature, restricted intervals are also referred to as actual, and unlimited as improper . ${\ displaystyle - \ infty}$${\ displaystyle \ infty}$

Infinite closed interval on the left
${\ displaystyle (- \ infty, b] = {] {- \ infty, b}]}: = \ {x \ in \ mathbb {R} \ mid x \ leq b \}}$

It contains all numbers that are less than or equal to. ${\ displaystyle b}$

Infinite open interval on the left
${\ displaystyle (- \ infty, b) = {] {- \ infty, b} [}: = \ {x \ in \ mathbb {R} \ mid x

It contains all numbers that are less than . ${\ displaystyle b}$

Infinite closed interval on the right
${\ displaystyle [a, \ infty) = {[{a, \ infty} [}: = \ {x \ in \ mathbb {R} \ mid a \ leq x \}}$

It contains all numbers that are greater than or equal to. ${\ displaystyle a}$

Infinite open interval on the right
${\ displaystyle (a, \ infty) = {] {a, \ infty} [}: = \ {x \ in \ mathbb {R} \ mid a

It contains all the numbers that are greater than . ${\ displaystyle a}$

Infinite open (and at the same time closed) interval on both sides
${\ displaystyle (- \ infty, \ infty) = {] {- \ infty, \ infty} [}: = \ mathbb {R}}$

It contains all the numbers between and . This corresponds to the entire set of real numbers ( ). ${\ displaystyle - \ infty}$${\ displaystyle + \ infty}$${\ displaystyle \ mathbb {R}}$

Incidentally, the above definition does not require that every interval is empty. In addition, depending on the application, there are also definitions that do not allow such intervals or, in the case of the case, simply swap the limits. ${\ displaystyle a \ leq b}$${\ displaystyle a> b}$${\ displaystyle a> b}$

To avoid confusion with the decimal point , the semicolon (;) is also used as a separator , rarely a vertical line (|), e.g. B.

${\ displaystyle (0,2 {,} 5] = (0; 2 {,} 5] = (0 | 2 {,} 5].}$

n -dimensional intervals

definition

Analogously, one defines any n -dimensional interval ( cuboid ) for in the n -dimensional space${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle \ mathbb {R} ^ {n}}$

${\ displaystyle I: = I_ {1} \ times \ dotsb \ times I_ {n}}$ at any intervals ${\ displaystyle I_ {1}, \ dotsc, I_ {n} \ subseteq \ mathbb {R}.}$

Bounded n -dimensional intervals

Let with and , then especially applies: ${\ displaystyle a, b \ in \ mathbb {R} ^ {n}}$${\ displaystyle a = (a_ {1}, \ dotsc, a_ {n})}$${\ displaystyle b = (b_ {1}, \ dotsc, b_ {n})}$

Closed interval
${\ displaystyle [a, b]: = \ {(x_ {1}, \ dotsc, x_ {n}) \ in \ mathbb {R} ^ {n} \ mid a_ {1} \ leq x_ {1} \ leq b_ {1}, \ dotsc, a_ {n} \ leq x_ {n} \ leq b_ {n} \}.}$
Open interval
${\ displaystyle (a, b) = {] {a, b} [}: = \ {(x_ {1}, \ dotsc, x_ {n}) \ in \ mathbb {R} ^ {n} \ mid a_ {1}
Half-open (more precisely right-open) interval
${\ displaystyle [a, b) = {[{a, b} [}: = \ {(x_ {1}, \ dotsc, x_ {n}) \ in \ mathbb {R} ^ {n} \ mid a_ {1} \ leq x_ {1}
Half-open (more precisely left-open) interval
${\ displaystyle (a, b] = {] {a, b}]}: = \ {(x_ {1}, \ dotsc, x_ {n}) \ in \ mathbb {R} ^ {n} \ mid a_ {1}

generalization

In topology , real intervals are examples of connected sets; in fact, a subset of the real numbers is connected if and only if it is an interval. Open intervals are open sets and closed intervals are closed sets. Half-open intervals are neither open nor closed. Closed bounded intervals are compact .

All notations made here for the real numbers can be transferred directly to any totally ordered sets. ${\ displaystyle \ mathbb {R}}$

literature

• Harro Heuser: Textbook of Analysis . Part 1. 5th edition. Teubner-Verlag, 1988, ISBN 3-519-42221-2 , p. 84

3. See, however, the closed intervals in the extended real numbers ${\ displaystyle {\ overline {\ mathbb {R}}}.}$