Find the word definition

Wiktionary
compact space

n. (context mathematics English) Any topological subset of Euclidean space that is a compact set

Wikipedia
Compact space

One such generalization is that a space is sequentially compact if any infinite sequence of points sampled from the space must frequently (infinitely often) get arbitrarily close to some point of the space. An equivalent definition is that every sequence of points must have an infinite subsequence that converges to some point of the space. The Heine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers accumulate to 0 (others accumulate to 1). The same set of points would not accumulate to any point of the open unit interval ; so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points has no subsequence that converges to any given real number.

Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions. The term compact was introduced into mathematics by Maurice Fréchet in 1904 as a distillation of this concept. Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation. A typical application is furnished by the Arzelà–Ascoli theorem or the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.

Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces. In general topological spaces, however, different notions of compactness are not necessarily equivalent. The most useful notion, which is the standard definition of the unqualified term compactness, is phrased in terms of the existence of finite families of open sets that " cover" the space in the sense that each point of the space must lie in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character.

The term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of a topological space.

Compact Space (band)

Compact Space is a band featuring Christian Eigner, Daryl Bamonte and vocalist Florian Kraemmer. They started writing together in 2008 and released their first album in 2011. The first single 'Push Push' was released on iTunes and Amazon in October 2009 through their own label Elsbeere Recordings, to great acclaim. On the internet, they also placed two other songs "Reverse Happiness" and "My Other Life" for people to stream on their MySpace page.

The album and single were produced by Christian Eigner and mixed by Paul PDub Walton. The artwork for the single is by Roger O'Donnell. In June 2011, the trio released their first full-length album 'Nameless' in Germany, Switzerland and Austria, in conjunction with Ferryhouse Publishing, a company based on Hamburg. In support of the release, an intimate gathering for 200 press and fans was organised at the Planetarium in Hamburg, Germany on 23 June. The album was played in full and accompanied by videos projected on the ceiling of the main theatre. In an interview from that evening Florian and Christian explained that their working relationship has existed for many years (Florian provided vocals on Eigner's 'Recovery' amongst other previous collaborations) and Daryl's lyrical contributions added a new dimension to their songwriting. The album is set for release worldwide later in 2011 or early 2012.

The digital "Push Push E.P." was also released on the same date. A second E.P. called "Who Says It's Real" with a new track and some mixes was released on 23 December.

Writing for a second album has commenced in August 2011, as posted on Daryl's Twitter Account

Usage examples of "compact space".

Hani ships going far and wide across Compact space with males aboard and all the attendant mindset in the crews.

He had been off a time fighting Akkhtimakt, trying to open the way for a human fleet now enroute to Compact space, and now Jik had made a deal which would effectively bring Sikkukkut into alliance with the mahendo'sat against Akkhtimakt, emphatically not the situation Goldtooth was working toward.

Humans were headed into Compact space in great number, and Goldtooth's whole plan for human-mahen alliance now was jeopardized by the taking of Kefk and its delivery to Sikkukkut, who consequently would bring the kif into unity under one hakkikt much faster then Goldtooth's plans called for.

She'd just sent a very dangerous mahen agent wandering through station computer records to ask himself why she'd bought what she'd bought, and why station life-support chemicals, basic foodstuffs, and exploding rocks nobody in Compact space had wanted to buy .

She'd just sent a very dangerous mahen agent wandering through station computer records to ask himself why she'd bought what she'd bought, and why station life-support chemicals, basic foodstuffs, and exploding rocks nobody in Compact space had wanted to buy.

He had been off a time fighting Akkhtimakt, trying to open the way for a human fleet now enroute to Compact space, and now Jik had made a deal which would effectively bring Sikkukkut into alliance with the mahendo’.

Humans were headed into Compact space in great number, and Goldtooth’.

The only illumination came from two electric lamps-quite inadequate to illuminate that long, compact space.

He took a seat at his desk, offering me the only other chair in the compact space.

But banks themselves were not without their compromised accesses, where stsho were concerned, since stsho had set up the banking system, all through Compact space: stsho technology, stsho procedures, stsho rules of accounting and the stsho system of transfers and debits.