Example: The blue circle represents the set of points (x, y) satisfying x2 + y2 = r2. The red disk represents the set of points (x, y) satisfying x2 + y2 < r2. The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set.
In mathematics, an open set is a generalization of an open interval in the real line.
In a metric space (a set along with a distance defined between any two points), an open set is a set that, along with every point P, contains all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).
More generally, an open set is a member of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, every subset can be open (the discrete topology), or no subset can be open except the space itself and the empty set (the indiscrete topology).[1]
In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such as continuity, connectedness, and compactness, which were originally defined by means of a distance.
The most common case of a topology without any distance is given by manifolds, which are topological spaces that, near each point, resemble an open set of a Euclidean space, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology, which is fundamental in algebraic geometry and scheme theory.
In mathematics, an openset is a generalization of an open interval in the real line. In a metric space (a set along with a distance defined between any...
In topology, a clopen set (a portmanteau of closed-openset) in a topological space is a set which is both open and closed. That this is possible may...
opensets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the opensets are the upper sets of...
{\displaystyle S} of a topological space X {\displaystyle X} is called a regular openset if it is equal to the interior of its closure; expressed symbolically,...
mathematics, a Borel set is any set in a topological space that can be formed from opensets (or, equivalently, from closed sets) through the operations...
mathematics, a closed set is a set whose complement is an openset. In a topological space, a closed set can be defined as a set which contains all its...
a topology, the most commonly used of which is the definition through opensets, which is easier than the others to manipulate. A topological space is...
more specifically in topology, an open map is a function between two topological spaces that maps opensets to opensets. That is, a function f : X → Y {\displaystyle...
dense sets need not contain any non-empty openset. The intersection of two dense open subsets of a topological space is again dense and open. The empty...
the concept of opensets. If we change the definition of 'openset', we change what continuous functions, compact sets, and connected sets are. Each choice...
topological space X, the empty set is open by definition, as is X. Since the complement of an openset is closed and the empty set and X are complements of...
(contains a dense openset), a comeagre set need not be a G δ {\displaystyle G_{\delta }} set (countable intersection of opensets), but contains a dense...
In fractal geometry, the openset condition (OSC) is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions...
charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic...
said to be disconnected if it is the union of two disjoint non-empty opensets. Otherwise, X {\displaystyle X} is said to be connected. A subset of a...
Baire), or is called an almost openset, if it differs from an openset by a meager set; that is, if there is an openset U ⊆ X {\displaystyle U\subseteq...
Openset, in mathematics Open interval, in mathematics Open line segment, in mathematics Open map, in mathematics Open (2011 film), a 2011 film Open (2019...
convex). This neighborhood can also be chosen to be an openset or, alternatively, a closed set. Let X {\displaystyle X} be a vector space over the field...
numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervals I {\displaystyle I} covers E {\displaystyle...
equal to the boundary of some openset (for example the openset can be taken as the complement of the set). An arbitrary set A ⊆ X {\displaystyle A\subseteq...
the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function...
specifications, open education, open educational resources, open government, open knowledge, open access, open science, and the open web. The growth of the open data...
dimensional. The Cantor ternary set C {\displaystyle {\mathcal {C}}} is created by iteratively deleting the open middle third from a set of line segments. One starts...
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