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Basic subset of a topological space
and mathematical analysis, 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
Open_set
Subset which is both open and closed
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
Clopen_set
Class of mathematical sets
open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is
Borel_set
{\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,
Regular_open_set
Open set containing a given point
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
Neighbourhood_(mathematics)
Mathematical set containing no elements
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
Empty_set
Collection of open sets used to define a topology
a family B {\displaystyle {\mathcal {B}}} of open subsets of X {\displaystyle X} such that every open set of the topology is equal to the union of some
Base_(topology)
Complement of an open subset
terms of its open sets, which determine what counts as a "neighborhood" of its points. A set is closed if it is the complement of an open set. In metric
Closed_set
Branch of topology
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
General_topology
Subset whose closure is the whole space
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
Dense_set
Condition for fractals in math
In fractal geometry, the open set condition (OSC) is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions
Open_set_condition
general topology, a saturated set is a subset of a topological space equal to an intersection of (an arbitrary number of) open sets. Let S {\displaystyle S}
Saturated set (intersection of open sets)
Saturated_set_(intersection_of_open_sets)
Topology on prime ideals and algebraic varieties
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
Zariski_topology
"Small" subset of a topological space
(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
Meagre_set
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
Glossary_of_general_topology
Branch of mathematics
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
Topology
Largest open subset of some given set
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
Interior_(topology)
Mathematical space with a notion of closeness
a topology, the most commonly used of which is the definition through open sets. A topological space is the most general type of a mathematical space
Topological_space
Algorithm used for pathfinding and graph traversal
while open_set is not empty // This operation can occur in O(Log(N)) time if open_set is a min-heap or a priority queue current := the node in open_set having
A*_search_algorithm
Topics referred to by the same term
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
Open
Functions that send open (resp. closed) subsets to open (resp. closed) subsets
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
Open_and_closed_maps
Collection of data
member of the data set. Data sets can also consist of a collection of documents or files. In the open data discipline, a data set is a unit used to measure
Data_set
Branch of mathematics that studies sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any
Set_theory
Region with boundary of finite measure
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
Caccioppoli_set
Countable intersection of open sets
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'
Gδ_set
Fractal sets in complex dynamics of mathematics
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
Julia_set
Openly accessible data
hardware, open content, open specifications, open education, open educational resources, open government, open knowledge, open access, open science, and
Open_data
Set of points on a line segment with certain topological properties
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
Cantor_set
Difference of an open set by a meager set
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
Property_of_Baire
Collection of mathematical objects
In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects:
Set_(mathematics)
Self-contained underwater breathing apparatus
of a rebreather dive is longer than an open-circuit dive, for similar weight and bulk of the set, if the set is bigger than the practical lower limit
Scuba_set
Invariant measure of fractal dimension
set A (in certain cases), we need a technical condition called the open set condition (OSC) on the sequence of contractions ψi. There is an open set V
Hausdorff_dimension
Subsets whose union equals the whole set
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
Cover_(topology)
Fractal named after mathematician Benoit Mandelbrot
The Mandelbrot set (/ˈmændəlbroʊt, -brɒt/) is a two-dimensional set. It is defined in the complex plane as the complex numbers c {\displaystyle c} for
Mandelbrot_set
Tool to track locally defined data attached to the open sets of a topological space
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
Sheaf_(mathematics)
Property of topological spaces
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
Compactly_generated_space
Comprehensive list of Magic: The Gathering card sets since its inception in 1993
The trading card game Magic: The Gathering has released a large number of sets since it was first published by Wizards of the Coast. After the 1993 release
List of Magic: The Gathering sets
List_of_Magic:_The_Gathering_sets
Broadest definition of sizes in integer-dimensional spaces
intersections, and complements. This includes open sets, closed sets, countable sets, intervals, boxes, and many other sets obtained from them by countable operations
Lebesgue_measure
Natural basic set in product spaces
Cylinder sets are clopen sets. As elements of the topology, cylinder sets are by definition open sets. The complement of an open set is a closed set, but
Cylinder_set
Topological space that is connected
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
Connected_space
Construct in functional analysis
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
Balanced_set
Set of the elements not in a given subset
In set theory, the complement of a set A, often denoted by A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the
Complement_(set_theory)
Linear operator in algebra and operator theory
resolvent set ρ ( L ) ⊆ C {\displaystyle \rho (L)\subseteq \mathbb {C} } of a bounded linear operator L is an open set. More generally, the resolvent set of
Resolvent_set
American artificial intelligence company
OpenAI is an American artificial intelligence (AI) research organization headquartered in San Francisco, consisting of OpenAI Group PBC, a for-profit
OpenAI
Mathematical set whose closure has empty interior
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
Nowhere_dense_set
All points and limit points in a subset of a topological space
are required to be open. The definition of a point of closure of a set is closely related to the definition of a limit point of a set. The difference between
Closure_(topology)
Golf tournament held in the United States
courses, the U.S. Open is set up so that scoring is very difficult, with a premium placed on accurate driving. As of 2026, the U.S. Open awards a $22.5 million
U.S._Open_(golf)
Concept in set theory
intersection is called a cylinder set, and the set of all such cylinder sets is a basis for the product topology. Every open set is the union of (a countable
Baire_space_(set_theory)
Point of a subset S around which there are no other points of S
an open ball around x that contains only finitely many elements of S. A point set that is made up only of isolated points is called a discrete set or
Isolated_point
Countable union of closed sets
The set R ∖ Q {\displaystyle \mathbb {R} \setminus \mathbb {Q} } of irrationals is not an Fσ set. In metrizable spaces, every open set is an Fσ set. The
Fσ_set
All points in the topological closure not belonging to the interior
Largest open set disjoint from some given set Interior (topology) – Largest open subset of some given set Nowhere dense set – Mathematical set whose closure
Boundary_(topology)
All numbers between two given numbers
interval is the empty set and does not depend on a {\displaystyle a} . The open intervals are those intervals that are open sets for the usual topology
Interval_(mathematics)
Type of function in mathematics
complex function on an open set is analytic if and only if it is holomorphic, that is, complex differentiable at every point of the set. For this reason, in
Analytic_function
Identities and relationships involving sets
mathematics, particularly in the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection
Algebra_of_sets
Index of articles associated with the same name
space Y is an open mapping Open mapping theorem (complex analysis), states that a non-constant holomorphic function on a connected open set in the complex
Open_mapping_theorem
Subset of a preorder that contains all larger elements
F} be the set of all (not-necessarily-open) neighborhoods of x {\displaystyle x} . Then F {\displaystyle F} is an upper set in the power set of X {\displaystyle
Upper_and_lower_sets
Vector space with a notion of nearness
closed sets need not be closed. The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need
Topological_vector_space
Type of play in football
The term set piece or set play is used in association football and rugby football to refer to a situation when the ball is returned to open play, for example
Set_piece_(football)
set Open set Clopen set Fσ set Gδ set Compact set Relatively compact set Regular open set, regular closed set Connected set Perfect set Meagre set Nowhere
List_of_types_of_sets
In mathematics, a concept that formalizes a certain idea of movement and mixing
discrete case, x ∈ X {\displaystyle x\in X} is non-wandering if, for every open set U containing x and every N > 0, there is some n > N such that μ ( f n (
Wandering_set
Hard-court tennis tournament
The US Open Tennis Championships, commonly called the US Open, is a hardcourt tennis tournament organized by the United States Tennis Association annually
US_Open_(tennis)
Mathematical set of all subsets of a set
mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed
Power_set
Mathematical measure for topological spaces
for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets. Let (X, T) be a topological
Regular_measure
Type of mathematical space
following basic open sets: every subset of N {\displaystyle \mathbb {N} } is open; the only open sets containing a are X and U; and the only open sets containing
Compact_space
Type of topological space
set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Given a set
Discrete_space
constructible set is a finite union of locally closed sets. (A set is locally closed if it is the intersection of an open set and closed set.) However, a
Constructible_set_(topology)
Subfield of mathematical logic
containing the open sets of X. This means that the Borel sets of X are the smallest collection of sets such that: Every open subset of X is a Borel set. If A is
Descriptive_set_theory
compact Gδ sets. In other words, the σ–algebra of Baire sets is the σ–algebra generated by all those intersections of countably many open sets that yield
Baire_set
Set of all things that may be the input of a mathematical function
non-empty connected open set in a topological space. In particular, in real and complex analysis, a domain is a non-empty connected open subset of the real
Domain_of_a_function
Mathematical set that can be enumerated
mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. Equivalently, a set is countable
Countable_set
Multiple equivalent ways to define a topological space
of topology, a topological space is usually defined by declaring its open sets. However, this is not necessary, as there are many equivalent axiomatic
Axiomatic foundations of topological spaces
Axiomatic_foundations_of_topological_spaces
Generalization of a sequence of points
collections of open sets in topological spaces are much like directed sets in behavior. For an example where sequences do not suffice, interpret the set R R {\displaystyle
Net_(mathematics)
Set of a ring's prime ideals
space; that is, commutative rings are associated to every point and every open set, which satisfy some compatibility conditions. The structure formed by the
Spectrum_of_a_ring
Set of elements common to all of some sets
In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing
Intersection_(set_theory)
Standard system of axiomatic set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in
Zermelo–Fraenkel_set_theory
nonempty open sets are disjoint. X cannot be written as the union of two proper closed subsets. Every nonempty open set is dense in X. Every open set is connected
Hyperconnected_space
Pattern-finding real-time card game
Set (stylized as SET or SET!) is a real-time card game designed by Marsha Falco in 1974 and published by Set Enterprises in 1991. The deck consists of
Set_(card_game)
Property in descriptive set theory
property, and can be written as the disjoint union of a perfect set and a countable open set. As a consequence, if a subset S ⊂ X {\displaystyle S\subset
Perfect_set_property
Topology where a set is open if it contains a particular point
is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and p ∈ X. The collection
Particular_point_topology
Amount of variation between extrema
function at a point, and oscillation of a function on an interval (or open set). Let ( a n ) {\displaystyle (a_{n})} be a sequence of real numbers. The
Oscillation_(mathematics)
Topology made of cocountable subsets
infinite set X {\displaystyle X} . In this topology, a set is open if its complement in X {\displaystyle X} is either countable or equal to the entire set. Equivalently
Cocountable_topology
Any one of the distinct objects that make up a set in set theory
mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four
Element_of_a_set
Golf tournament
Pebble Beach Golf Links The 2000 United States Open Championship was the 100th U.S. Open Championship, held June 15–18 at Pebble Beach Golf Links in Pebble
2000_U.S._Open_(golf)
Any collection of sets, or subsets of a set
family of sets (whose elements are called open sets) over X {\displaystyle X} that contains both the empty set ∅ {\displaystyle \varnothing } and X {\displaystyle
Family_of_sets
Tennis championship
second to lose all of his first three Australian Open finals, after Andy Murray. Medvedev also set Open Era records for the most time spent playing at one
2024 Australian Open – Men's singles
2024_Australian_Open_–_Men's_singles
Topological space in which the closure of every open set is open
disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word
Extremally_disconnected_space
Annual tennis tournament held in Melbourne
Plexicushion (2008–2019), and blue GreenSet since 2020. First held in 1905 as the Australasian championships, the Australian Open has grown to become one of the
Australian_Open
Inherited topology
{\displaystyle S} is open in the subspace topology if and only if it is the intersection of S {\displaystyle S} with an open set in ( X , τ ) {\displaystyle
Subspace_topology
British zombie horror miniseries
Dead Set is a British satirical zombie comedy horror television miniseries created and written by Charlie Brooker and directed by Yann Demange. Set on the
Dead_Set
Intersection of an open set and a closed set
are satisfied: E {\displaystyle E} is the intersection of an open set and a closed set in X . {\displaystyle X.} For each point x ∈ E , {\displaystyle
Locally_closed_subset
Generalized topological space
dropping the requirements that the set of open sets be closed under union and finite intersection, that the open sets be extensional, and that the membership
Chu_space
Tennis tournament
record for most wins (two). 2010 ATP tournament profile "ATP Valencia Open set to sell their tournament status and downgrade". 7 February 2015. "Scoreboard:
Valencia_Open
Degree of differentiability of a function or map
made for functions defined on open subsets of Euclidean space. For functions on closed intervals, closures of open sets, or more general subsets, the
Smoothness
Theorem on the equality of analytic functions
example, if D consists of two disjoint open sets, f {\displaystyle f} can be 0 {\displaystyle 0} on one open set, and 1 {\displaystyle 1} on another, while
Identity_theorem
Type of continuous map in topology
{\displaystyle x_{1}>0} , the set U := { ( x 1 , x 2 ) ∈ S 1 ∣ x 1 > 0 } {\displaystyle U:=\{(x_{1},x_{2})\in S^{1}\mid x_{1}>0\}} is an open neighborhood of x {\displaystyle
Covering_space
Binary sequence
measures of any such sequence. An effective open set is an open set that is the union of the sequence of basic open sets determined by a recursively enumerable
Algorithmically random sequence
Algorithmically_random_sequence
Mathematical set containing all objects
In set theory, a universal set is a set that contains all of the objects in the theory, including itself. In set theory as usually formulated, it can
Universal_set
American rock band
Summer Set Open Up About 'Chelsea' (Kane)". Jsyk.com. Archived from the original on July 21, 2011. Retrieved August 3, 2011. "The Summer Set's Taylor
The_Summer_Set
Relationship between certain categories
function f is continuous if the inverse image f −1(O) of any open set in the codomain of f is open in the domain of f. Thus any continuous function f from
Stone_duality
Definition of continuity for functions between posets
functions of open sets, and thus Sierpiński space is the classifying space for open sets. A subset O of a partially ordered set P is called Scott-open if it
Scott_continuity
OPEN SET
OPEN SET
Male
Welsh
 Modern Welsh form of Old Welsh Owain, OWEN means "born of yew." Compare with another form of Owen.
Boy/Male
English
Open.
Boy/Male
English French
Open.
Boy/Male
Celtic Welsh
Son of Owen.
Boy/Male
English French
Open.
Female
Thai/Siamese
Thai name PEN-CHAN means "full moon."
Boy/Male
English French American
Open.
Male
Welsh
Variant form of Welsh Owen, possibly OUEN means "born of yew."
Boy/Male
English French
Open.
Boy/Male
English French
Open.
Surname or Lastname
English
English : variant of Penn.Dutch : metonymic occupational name for a clerk or penman, from Dutch pen ‘pen’.Cambodian : unexplained.
Boy/Male
Welsh
Son of Owen.
Boy/Male
English French
Open.
Boy/Male
Hindu, Indian
Open
Female
English
English short form of Latin Penelope, PEN means "weaver of cunning."
Boy/Male
American, British, English, French
Open; Variant of Darrel Open
Male
English
 Anglicized form of Irish Gaelic Eóghan, OWEN means "born of yew." Compare with another form of Owen.
Male
Swedish
Norwegian and Swedish form of Old Norse Óðinn, ODEN means "poetry, song" and "eager, frenzied, raging."
Boy/Male
English French
Open.
Boy/Male
French
Open.
OPEN SET
OPEN SET
Boy/Male
Irish Hebrew
Servant.
Girl/Female
Indian, Punjabi, Sikh
A Light of God
Boy/Male
British, English
Wealthy Wolf
Girl/Female
Hindu, Indian
Success
Girl/Female
Tamil
Ashrika | à®…à®·à¯à®°à¯€à®•à®¾
Someone gives shelter
Girl/Female
Indian
An Angel
Boy/Male
Indian
Name of prophets sword
Boy/Male
Muslim
Very clever
Boy/Male
Hindu
Sage
Boy/Male
Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Worshipped
OPEN SET
OPEN SET
OPEN SET
OPEN SET
OPEN SET
a.
Uttered with a relatively wide opening of the articulating organs; -- said of vowels; as, the an far is open as compared with the a in say.
n.
Open or unobstructed space; clear land, without trees or obstructions; open ocean; open water.
a.
With eyes widely open; watchful; vigilant.
a.
Not of a quality to prevent communication, as by closing water ways, blocking roads, etc.; hence, not frosty or inclement; mild; -- used of the weather or the climate; as, an open season; an open winter.
a.
Taking place in the open air; outdoor; as, an open-air game or meeting.
a.
Having the mouth open; gaping; hence, greedy; clamorous.
a.
Free; disengaged; unappropriated; as, to keep a day open for any purpose; to be open for an engagement.
v. t.
To enter upon; to begin; as, to open a discussion; to open fire upon an enemy; to open trade, or correspondence; to open a case in court, or a meeting.
a.
Open.
v. t.
To spread; to expand; as, to open the hand.
a.
Not drawn together, closed, or contracted; extended; expanded; as, an open hand; open arms; an open flower; an open prospect.
v. t. & i.
To open.
a.
Free to be used, enjoyed, visited, or the like; not private; public; unrestricted in use; as, an open library, museum, court, or other assembly; liable to the approach, trespass, or attack of any one; unprotected; exposed.
v. t.
To make or set open; to render free of access; to unclose; to unbar; to unlock; to remove any fastening or covering from; as, to open a door; to open a box; to open a room; to open a letter.
a.
Produced by an open string; as, an open tone.
v. t.
To loosen or make less compact; as, to open matted cotton by separating the fibers.
a.
Not settled or adjusted; not decided or determined; not closed or withdrawn from consideration; as, an open account; an open question; to keep an offer or opportunity open.
a.
Free of access; not shut up; not closed; affording unobstructed ingress or egress; not impeding or preventing passage; not locked up or covered over; -- applied to passageways; as, an open door, window, road, etc.; also, to inclosed structures or objects; as, open houses, boxes, baskets, bottles, etc.; also, to means of communication or approach by water or land; as, an open harbor or roadstead.
a.
Free or cleared of obstruction to progress or to view; accessible; as, an open tract; the open sea.
a.
Not concealed or secret; not hidden or disguised; exposed to view or to knowledge; revealed; apparent; as, open schemes or plans; open shame or guilt.