Dense-in-itself

In mathematics, a subset A of a topological space is said to be dense-in-itself if A contains no isolated points.

Every dense-in-itself closed set is perfect. Conversely, every perfect set is dense-in-itself.

A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers. This set is dense-in-itself because every neighborhood of an irrational number x contains at least one other irrational number $y \neq x$. On the other hand, this set of irrationals is not closed because every rational number lies in its closure. For similar reasons, the set of rational numbers is also dense-in-itself but not closed.

See also

• Dense set
• Nowhere dense set
• Dense order
• Perfect space

This article incorporates material from Dense in-itself on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Topology Geometry stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем решить контрольную работу Hot House Flowers Orange Dove Look at other dictionaries: dense-in-itself — adjective Having no isolated points …   Wiktionary Dense set — In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point x in X belongs to A or is a limit point of A.[1] Informally, for every point in X, the point is either in A or arbitrarily close …   Wikipedia Dense order — In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x < y, there is a z in X such that x < z < y. The rational numbers with the ordinary ordering are a densely ordered set in this sense, as …   Wikipedia Dense heterarchy — Hierarchical organization in social insect colonies can be thought of as a dense heterarchy in which the higher levels affect the lower levels and lower levels eventually influence the higher levels. Individual ants within the colony network are… …   Wikipedia Nowhere dense set — A subset A of a topological space X is nowhere dense in X if and only if the interior of the closure of A is empty. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure …   Wikipedia Big Bang — This article is about the cosmological model. For the sitcom, see The Big Bang Theory. For other uses, see Big Bang (disambiguation) …   Wikipedia Density (disambiguation) — Density and dense usually refer to a measure of how much of some entity is within a fixed amount of space. Types of density include: In physics, density of mass: Density, mass per volume Area density or surface density, mass over a (two… …   Wikipedia List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… …   Wikipedia Isolated point — 0 is an isolated point of A In topology, a branch of mathematics, a point x of a set S is called an isolated point of S, if there exists a neighborhood of x not containing other points of S. In particular, in a Euclidean space (or in a …   Wikipedia Vertical draft — For the band, see Air Pocket (band) An updraft or downdraft is the vertical movement of air as a weather related phenomenon. One of two forces causes the air to move. Localized regions of warm or cool air will exhibit vertical movement. A mass of …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”