The "'interior "'of an interval is the largest open interval that is contained in; it is also the set of points in which are not endpoints of.
42.
It consists of the product of two copies of the Sorgenfrey line, which is the real line \ mathbb { R } under the half-open interval topology.
43.
For example, the open interval on the integers is empty since there are no integers " i " such that 1 < " i " < 2.
44.
Note that the theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.
45.
Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e . g ., the open interval ( 0, 1 ) is Polish.
46.
The theorem cannot be applied to this function, clearly, because it does not satisfy the condition that the function must be differentiable for every x in the open interval.
47.
For example, let " X " be the set of ordinals at most equal to the first uncountable ordinal ?, with the topology generated by " open intervals ".
48.
Therefore, the polynomial p ( x ) has either two or no real roots in the open interval ( 0, 2 ), a case that must be further investigated.
49.
Therefore, the polynomial p ( x ) has either two or no real roots in the open interval ( 0, 2 ), a case which should be further investigated.
50.
A function that is equal to its Taylor series in an open interval ( or a disc in the complex plane ) is known as an analytic function in that interval.
