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...

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...

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...

{\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,...

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

cover of a set is a subset of the cover that also covers the set. A cover is called an open cover if each of its elements is an open set. Covers are...

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...

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...

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...

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...

(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...

topological space X, the empty set is open by definition, as is X. Since the complement of an open set is closed and the empty set and X are complements of...

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...

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...

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

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...

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...

closed sets in the real line. Intervals ( a , b ] {\displaystyle (a,b]} and [ a , b ) {\displaystyle [a,b)} are neither an open set nor a closed set. If...

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...

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...

Open set, in mathematics Open interval, in mathematics Open line segment, in mathematics Open map, in mathematics Open (2011 film), a 2011 film Open (2019...

Largest open set disjoint from some given set Interior (topology) – Largest open subset of some given set Nowhere dense set – Mathematical set whose closure...

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'...

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...

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...

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 open set (for example the open set can be taken as the complement of the set). An arbitrary set A ⊆ X {\displaystyle A\subseteq...

specifications, open education, open educational resources, open government, open knowledge, open access, open science, and the open web. The growth of the open data...

convex). This neighborhood can also be chosen to be an open set or, alternatively, a closed set. Let X {\displaystyle X} be a vector space over the field...