If S is compact but not closed, then it has an accumulation point a not in S. Consider a collection consisting of an open neighborhood N ( x ) for each x ∈ S, chosen small enough to not intersect some neighborhood V < sub > x </ sub > of a.

Consider the open balls centered upon a common point, with any radius.

Consider an open subset U of the complex plane C. Let a be an element of U, and f: U

Consider for instance the open unit disc, a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss – Bonnet formula does not work.

Consider an open set on the real line and a function f defined on that set with real values.

Consider a scheme X and a cover by affine open subschemes Spec A < sub > i </ sub >.

Consider too, with what force, diligence and vivacity he has rendered back all this which in Johnson's neighbourhood, his " open sense " had so eagerly and freely taken in.

Consider adopting a system of holidays in which time off is granted with an eye to minimum inconvenience to the operation of the plant.

Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by Af.

( Consider 1 / 0, which is defined with the value of infinity, vs. 0 / 0, which is undefined.

Consider a car's cruise control, which is a device designed to maintain vehicle speed at a constant desired or reference speed provided by the driver.

Consider Japan, for instance, which used to have optional jury trials for capital or other serious crimes between 1928 and 1943.

Consider for example determining which of the following are to be considered diseases ( i. e., abnormal states requiring cure ): alcoholism, homosexuality, and chronic fatigue syndrome.

Consider the case in which liquid crystal molecules are aligned parallel to the surface and an electric field is applied perpendicular to the cell as in the following diagram.

Consider a binary electrolyte AB which dissociates into A + and B-ions and the equilibrium state is represented by the equation:

Consider a hypothetical event which has 8 possible outcomes, in a country using a decimal currency such as dollars.

Consider two ISPs, A and B, which each have a presence in New York, connected by a fast link with latency 5 ms ; and which each have a presence in London connected by a 5 ms link.

Consider the analogy between the sentence " Snow is white " and the character named Snow White, both of which can be true in some sense.

Consider a system in which voters can vote for any candidate from any one of many parties ; suppose further that if a party gets 15 % of votes, then that party will win 15 % of the seats in the legislature.

Consider a traveling transverse wave ( which may be a pulse ) on a string ( the medium ).

Consider Hawaiian, which has a and, but no or.

Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest.

Consider a customer purchasing an item which costs five pence, who possesses several silver sixpence coins.

Consider a typical rational number 415 / 93, which is around 4. 4624.

Consider the implication " if I am in Massachusetts, then I am in North America ", which we might alternatively express as, " if I were in Massachusetts, then I would be in North America ".

Consider an operation like vector addition, which is used very often in computer graphics applications.

Consider the problem of determining the index of the database entry which satisfies some search criterion.

As a sidenote, the fact that artificial intelligences are accepted as citizens of the Culture was a major factor in the Idiran-Culture War, which is explored in Consider Phlebas.

Consider the following claim: " it is false that A and B are both true ", which is written as:

Consider the direct system composed of the groups Z / p < sup > n </ sup > Z and the homomorphisms Z / p < sup > n </ sup > Z → Z / p < sup > n + 1 </ sup > Z which are induced by multiplication by p. The direct limit of this system consists of all the roots of unity of order some power of p, and is called the Prüfer group Z ( p < sup >∞</ sup >).

Consider climbing up the connectivity ladder — assume X is a simply-connected CW-complex whose 0-skeleton consists of a point.

Consider a steady flow in which its boundary consists of a streamline.

Consider a received discrete signal,, of independent samples that consists of an unknown constant with additive white Gaussian noise ( AWGN ) with known variance ( i. e., ).

To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use étale coverings instead.

The Lebesgue covering dimension is defined as the minimum number of overlaps that small open balls need to have in order to completely cover the object.

When you try to cover a line by dropping open intervals on it, you always end up covering some points twice.

While the Conditions of Carriage require period Travelcard holders to touch in and touch out at the start and end of their journey ( which is usually necessary to open automatic ticket gates ), Oyster card users with a valid period Travelcard covering their entire journey are not liable to pay a penalty fare when they have not touched in.

A Chinese martial arts greeting features the right hand in a fist with the left hand open covering it with a slight nod of the head.

Although the whistle is essentially a diatonic instrument, it is possible to get notes outside the principal major key of the whistle, either by half-holing ( partially covering the highest open finger hole ) or by cross-fingering ( covering some holes while leaving some higher ones open ).

Upon hatching, the chicks are semi-precocial, having open eyes, a dense covering of white or grey down feathers, and the ability to move around the nesting site.

Metz and its open countries are included into the forest and crop Lorraine Regional Natural Park, covering a total area of.

Before loading, a piece of the paper or canvas covering had to be cut open immediately under the vent ; after the shot had been rammed home the vent was filled with powder from a priming horn, and the gun was then fired by means of a hot iron, quick match or port-fire.

The Saigon press corps officially protest Nhu's " open physical intimidation to prevent the covering of news which we feel Americans have a right to know ".

It is a unique ecological system covering more than and a mosaic of creeks, open water, perennial marshes, seasonal wetlands, riparian forests, oak woodlands and grasslands ... As the receiving water of a watershed where most of the county's human population lives, it is a landscape feature of critical importance to Sonoma County's water quality, flood control, and biodiversity.

It can also affect and erode, through the enzymes it produces, the thin bone structure that isolates the top of the ear from the brain, as well as lay the covering of the brain open to infection with serious complications ( rarely even death due to causing a brain abscess and septicemia ).

Indeed, the Heine – Borel definition of compactness — that every covering of a space by open sets admits a finite subcovering — is relatively recent.

This is commonly seen in stone construction, where columns ( piles ) are used to support spans ( posts ) covering open spaces.

They began a new large-scale open pit operation covering.

Shoreview has an extensive park system consisting of 15 parks covering 1, 400 acres of parkland and open spaces.

Work continued, with phase one, covering the section from Evesham to Bidford Bridge, being declared open on 12 June 1971, during an IWA boat rally at Bidford.

Opencast mining finally resumed ten years later, after the site had been pumped out, but the coal reserves are now exhausted, and the site forms a nature reserve, with wet grasslands, reedbeds and open water covering.

An additional benefit of using containers to hold the body is that if the soil covering the corpse is washed away by a flood or some other natural process, the corpse will still not be exposed to open air.

The second largest open cast mine in the world, covering nearly 30 km² and up to 400m deep.

The range of this species is quite large, covering all of mainland India up to the Himalayas, and southeasterly in a broad band into Burma ( Myanmar ), Laos, and Thailand in open forest consisting of scrub, plantations and gardens.

These were successful at covering the Austrian rear, keeping the bridges over the Elbe open for retreating Austrian soldiers, and preventing pursuit by the Prussians, but at a terrible cost: 2, 000 men and almost as many horses were killed, wounded or captured in the action.