Interval (mathematics)
As an interval of the order topology and related areas of mathematics is called a " coherent " subset of a total ( or linear) child support amount in the Analysis. The interval consists of all elements that can be compared to two limiting elements of the interval, the lower and the upper limit of the interval in accordance with the size and that are within the meaning of this comparison between the limits. The limits of the interval may belong to the interval ( closed interval, ), do not belong ( open interval ) or partially belong to ( semi-open interval; ).
Contiguous means: If two objects are included in the subset, then so are all the objects ( in the amount of support ) in between, included. The most important examples for carrier quantities, the amounts of real, rational, and the whole of the natural numbers. In the above cases, and more generally, whenever a difference between two elements of the carrier set is explained, refers to the difference between the upper and lower limit of the interval () as the length of the interval or short interval length.
- 3.1 Restricted intervals
Examples
In this case, a discrete amount of the elements are adjacent to the interval.
The set of numbers between 0 and 1, wherein the end points of 0 and 1 are included.
Trivial examples of intervals are the empty set and quantities that contain exactly one element. If you do not want to include these, then one speaks of real intervals.
The amount may also be considered as a subset of the carrier set of real numbers. In this case, it is not by an interval, as the amount, for example, does not contain the past 6-7 non-natural numbers.
The carrier set of real numbers plays a special role in this respect under the said carrier amounts for intervals, as it is order complete (see a Dedekind cut ). Intervals in this case are precisely the coherent in the sense of topology subsets.
Designations and notations
An interval can be (both sides) or limited - even unilaterally - unlimited. It is uniquely determined by its lower and its upper bound of the interval, if is also indicated whether these limits are contained in the interval.
There are two different interval notation commonly used. In the more common of the two is used for boundaries which belong to the interval brackets and round for limits that do not belong to the interval. In the other angular notation used in place of the parentheses outwardly -turned ( reflected ). Below, both spellings are shown and compared with the set notation:
Restricted intervals
A limited interval is complete when it contains both limits, and open when both boundaries are not included. A limited interval is called semi-open if it contains exactly one of the two interval limits.
The interval contains both.
An interval is compact if it is closed and bounded.
The interval contains neither.
The interval includes, but is not.
The interval contains not, but probably.
In the case of and is the open unit interval and the closed unit interval.
Unlimited intervals
If the interval border missing on one side, so it there should be no barrier, is called a (on this page ) unbounded interval. Most of this, the known symbols and as a "substitute " interval boundaries are used, which itself never belong to the interval (hence the spelling of a round bracket ). In some literature limited intervals are referred to as actually unlimited as inauthentic.
It contains all the numbers that are smaller or equal.
It contains all the numbers which are smaller than.
It contains all the numbers which are equal to or greater.
It contains all numbers that are greater than.
It contains all the numbers between and. This corresponds to the real numbers ().
In the above definition is not the way required, so that for each interval is empty. There also exist depending on the application definitions that do not allow such intervals or in the case just switch the limits.
To avoid confusion with the decimal point and the semicolon ( ;), rarely, a vertical line (used as the delimiter | ) is used, for example,
N-dimensional intervals
Analogue is defined in the n- dimensional space for any n-dimensional interval (box )
Restricted intervals
There are now using and then applies specifically:
Generalization
In the topology of real intervals are examples of connected sets, is actually a subset of the real numbers even if and only connected if it is an interval. Open intervals are open sets and closed intervals are closed sets. Semi-open sets 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.