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.

The open interval is not compact: the open cover for does not have a finite subcover.

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 ) 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 ).

A manic episode is defined in the American Psychiatric Association's diagnostic manual as a period of seven or more days ( or any period if admission to hospital is required ) of unusually and continuously effusive and open elated or irritable mood, where the mood is not caused by drugs / medication or a medical illness ( e. g., hyperthyroidism ), and ( a ) is causing obvious difficulties at work or in social relationships and activities, or ( b ) requires admission to hospital to protect the person or others, or ( c ) the person is suffering psychosis.

To qualify for this income tax deduction, the easement must be: a ) perpetual ; b ) held by a qualified governmental or non-profit organization ; and, c ) serve a valid " conservation purpose ", meaning the property must have an appreciable natural, scenic, historic, scientific, recreational, or open space value.

We will redo the record, open up the doors for it to get on the r & b charts and make the black stations to play the record ...

:: For any open neighborhood N of A, there is a positive constant T such that f ( t, b ) ∈ N for all real t > T.

a and x < sub > n </ sub > = b. There are two types of Newton – Cotes formulae, the " closed " type which uses the function value at all points, and the " open " type which does not use the function values at the endpoints.

::( b ) offices and positions must be open to everyone under conditions of fair equality of opportunity

::( b ) attached to offices and positions open to all under conditions of fair equality of opportunity.

Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.

Robert Bruce Keanan Brown ( aka b < sup > 2 </ sup >) ( Editor 1988-1990 ) resigned in an open letter in Tertangala citing the " bureaucratisation " of the magazine as his main reason for leaving.

Unlike the virelai, the two " b " lines usually have exactly the same music and only in later ballate pick up the ( formerly distinctly French ) first and second ( open and close ) endings.

In 1968 there were a ) an amateur circuit including the Davis Cup (" closed " to any " contract " professional until 1973 ) and the Australian Championships, b ) two pro circuits: the " World Championship of Tennis ( WCT )" circuit and the " National Tennis League ( NTL )" circuit which met on 4 tournaments, and c ) an open circuit ( with a little more than 10 tournaments ).

All Fijian citizens aged 21 or over, " or such other age as the Parliament prescribes ," who have been physically resident in Fiji for the two years immediately prior to their registration, are entitled to enroll on ( a ) a Common Roll, as an elector for one of the 25 " open electorates ," so-called because they are open to candidates of all races, elected by voters of all races, and ( b ) one of the four communal ethnic rolls, as an elector for a constituency reserved for one of the four recognized ethnic groups.

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.