For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

In this case, is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns for every interval is sometimes called " the " Borel measure on R. In practice, even " the " Borel measure is not the most useful measure defined on the σ-algebra of Borel sets ; indeed, the Lebesgue measure is an extension of " the " Borel measure which possesses the crucial property that it is a complete measure ( unlike the Borel measure ).

) Note that the same set of points would not have, as an accumulation point, any point of the open unit interval ; hence that space cannot be compact.

The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself: an open ( or half-open ) interval of the real numbers is not compact.

In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it.

This subset I is referred to as the domain of f. Possible choices include I = R, the whole set of real numbers, an open interval

One starts by deleting the open middle third (, ) from the interval, leaving two line segments:

Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (- π, π ) on the x-axis ; then bend the ends of this interval upwards ( in positive y-direction ) and move them towards each other, until you get a circle with one point ( the topmost one ) missing.

The open interval ( 0, 1 ), again with the absolute value metric, is not complete either.

An example is given by the real numbers, which are complete but homeomorphic to the open interval ( 0, 1 ), which is not complete.

* The open interval ( a, b ) is homeomorphic to the real numbers R for any a < b. ( In this case, the bicontinuous forward mapping is given by and the inverse mapping is given by a scaled version of the function ).

The open interval ( a, b ) has the same measure, since the difference between the two sets consists only of the end points a and b and has measure zero.

Using the corresponding strict relation "<", one can also define the interval ( a, b ) as the set of points x satisfying a < x and x < b, also written a < x < b. An open interval may be empty even if a < b.

After liquid paint is applied, there is an interval during which it can be blended with additional painted regions ( at the " wet edge ") called " open time.

* The two points a and b form the boundary of the open interval.

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.

A standard example of such a system of neighbourhoods is for the real line R, where a subset N of R is defined to be a neighbourhood of a real number x if there is an open interval containing x and contained in N.

The standard ordering ≤ of the positive real numbers is not a well-ordering, since, for example, the open interval ( 0, 1 ) does not contain a least element.

* Some sets are neither open nor closed, for instance the half-open interval < nowiki >

If f is a differentiable function on R ( or an open interval ) and x is a local maximum or a local minimum of f, then the derivative of f at x is zero ; points where are called critical points or stationary points ( and the value of f at x is called a critical value ).

Here in these little rooms -- or stages arched open to the sky and river -- they choose a few lines out of the hundreds they may know and sing them according to one of the modes into which Persian music is divided.

The only chance of renovation is to open our eyes and see the mess.

For the occasion on which everyone already knows everyone else and the host wishes them to meet one or a few honored newcomers, then the `` open house '' system is advantageous because the honored guests are fixed connective points and the drifting guests make and break connections at the door.

All we want from Dr. Huxley's statement is the feeling that this is an open world, in the view of the best scientific opinion, with practically no directional commitments as to what may happen next, and no important confinements with respect to what may be possible.

Although open to the general public it is not overcrowded ; ;

On April 11th he wrote an open letter in The Advocate, making it known `` to the world that Jas. W. Robinson is by his own admission a base liar and a slanderer ''.

For there is also the `` face of reality '' in the form of the individual's perceptions of his own abilities and interests, of the objective possibilities open to him, of the familial and other social pressures to which he is exposed.

and the atmosphere is so befouled that no one dares walk in the open without respirators or soot plugs.

When all else fails, it is argued that open sessions slow down governmental operations.

But Interior Secretary Udall warns that there is a race on between those who would develop our few surviving open shorelines and those who would save them for the enjoyment of all as public preserves.

It seems that open season upon veterans' hospitalization is once more upon us.

There the truth is, open to eyes that are willing to look.

`` His mouth is open ''.

Experiments have shown that the shutter is 75 percent open in about 60-80 microseconds.

Ventilation is provided in a concrete block basement shelter by vents in the wall and by the open entrance.

The Glee Club is open to all students and faculty with no auditions necessary for membership.

The workshop is open five afternoons and two evenings each week.

A student organization, Bottega, is open to any student interested in increasing his understanding and appreciation of the graphic and ceramic arts in their historical, technical, and productive contexts.

Membership is open to any woman student in the College.

The effect of the recording is very open and natural, with the frequency emphasis exactly what you would expect from a live performance.

The single-barrel Stevens 940Y ( under $35 ) is made with a side lever rather than a top-tang lever because many youngsters aren't strong enough to operate a top tang to open a gun -- and the side lever does indeed open very easily.

The extent and location of open areas is noted.