In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two...

{\displaystyle S} of a topological space X {\displaystyle X} is called a regular open set if it is equal to the interior of its closure; expressed symbolically,...

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may...

mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations...

mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its...

open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of...

closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where...

(contains a dense open set), a comeagre set need not be a G δ {\displaystyle G_{\delta }} set (countable intersection of open sets), but contains a dense...

the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice...

(X, τ) is a family B {\displaystyle {\mathcal {B}}} of open subsets of X such that every open set of the topology is equal to the union of some sub-family...

the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories...

dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open. The empty...

more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X → Y {\displaystyle...

every set is open, every set is equal to its interior. In any indiscrete space X , {\displaystyle X,} since the only open sets are the empty set and X...

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...

In fractal geometry, the open set condition (OSC) is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions...

a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is...

X} . The cover C {\displaystyle C} is said to be an open cover if each of its members is an open set. That is, each U α {\displaystyle U_{\alpha }} is contained...

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any...

is open). A subset of X may be open, closed, both (a clopen set), or neither. The empty set and X itself are always both closed and open. An open subset...

sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set...

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...

Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U ⊆ X {\displaystyle U\subseteq...

zero-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...

set was defined as a functional, precisely a set function, for the first time: also, being defined on open sets, it can be defined on all Borel sets and...

said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, X {\displaystyle X} is said to be connected. A subset of a...

set is a subset of a topological space that is a countable intersection of open sets. The notation originated from the German nouns Gebiet 'open set'...

object, for each member of the data set. Data sets can also consist of a collection of documents or files. In the open data discipline, a dataset is a unit...

(estimated) cost nodes to expand. This priority queue is known as the open set, fringe or frontier. At each step of the algorithm, the node with the lowest...

every set A ⊆ X , {\displaystyle A\subseteq X,} A {\displaystyle A} is open in X {\displaystyle X} if and only if A ∩ K {\displaystyle A\cap K} is open in...