If a and b are any two real numbers such that a is less than b (a<b). Then the set {x€ R/ a≤ x ≤ b} is called a closed interval with end points a and b and is denoted by [a,b]. The set {x € R/a<x<b} is called an open interval with end points a and b and is denoted by (a,b).
Thus,
[a,b] = {x€ R/ a≤ x ≤ b} and(a,b) = {x € R/a<x<b}
Similarly we can define some other intervals as follows.
(a,b] = {x€ R/ a< x ≤ b}
[a,b) = {x€ R/ a≤ x < b}In these 4 intervals, the length of the interval is finite which is defined by b-a.
[a, ∞) = {x€ R/ x≥a}(a, ∞) = {x€ R/ x>a}
(-∞,a] = {x€ R/ x≤a}
(-∞,a) = {x€ R/ x<a}
In these 4 intervals, we cannot define the length and hence these 4 intervals are of infinite length.
Neighbourhoods:
Let a be any real number. If p>0 then the open interval (a-p, a+p) is called the p-neighbourhood of a. The following figure shows the location of (a-p, a+p) on the number line.
The set obtained by deleting a from this neighbourhood is called the deleted p-neighbourhood of a.i.e. the deleted p-neighbourhood of a is
(a,p, a) U (a,a+p) or (a-p, a+p) –{a}.
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