Open set

 ( General Topology ) 

In general topology 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 every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ).

More generally, an open set is a member of a given collection of Subset 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).

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, connected space, and compactness, which were originally defined by means of a distance.

The most common case of a topology without any distance is given by , 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.

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Motivation Intuitively, an open set provides a method to distinguish two points. For example, if about one of two points in a topological space, there exists an open set not containing the other (distinct) point, the two points are referred to as topologically distinguishable. In this manner, one may speak of whether two points, or more generally two , of a topological space are "near" without concretely defining a distance. Therefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are called .

In the set of all , one has the natural Euclidean metric; that is, a function which measures the distance between two real numbers: . Therefore, given a real number x, one can speak of the set of all points close to that real number; that is, within ε of x. In essence, points within ε of x approximate x to an accuracy of degree ε. Note that ε > 0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x = 0 and ε = 1, the points within ε of x are precisely the points of the interval (−1, 1); that is, the set of all real numbers between −1 and 1. However, with ε = 0.5, the points within ε of x are precisely the points of (−0.5, 0.5). Clearly, these points approximate x to a greater degree of accuracy than when ε = 1.

The previous discussion shows, for the case x = 0, that one may approximate x to higher and higher degrees of accuracy by defining ε to be smaller and smaller. In particular, sets of the form (− ε, ε) give us a lot of information about points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (− ε, ε)), one may find different results regarding the distance between 0 and other real numbers. For example, if we were to define as the only such set for "measuring distance", all points would be close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of . Thus, we find that in some sense, every real number is distance 0 away from 0. It may help in this case to think of the measure as being a binary condition: all things in are equally close to 0, while any item that is not in is not close to 0.

In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set ( X); rather than just the real numbers. In this case, given a point ( x) of that set, one may define a collection of sets "around" (that is, containing) x, used to approximate x. Of course, this collection would have to satisfy certain properties (known as axioms) for otherwise we may not have a well-defined method to measure distance. For example, every point in X should approximate x to some degree of accuracy. Thus X should be in this family. Once we begin to define "smaller" sets containing x, we tend to approximate x to a greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets about x is required to satisfy.

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Definitions Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one.

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Euclidean space A subset $U$ of the Euclidean space is open if, for every point in $U$, there exists a positive real number (depending on ) such that any point in whose Euclidean distance from is smaller than belongs to $U$.

(2026). 9780821832844, American Mathematical Society. ISBN 9780821832844

Equivalently, a subset $U$ of is open if every point in $U$ is the center of an open ball contained in $U.$

An example of a subset of that is not open is the closed interval , since neither nor belongs to for any , no matter how small.

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Metric space A subset U of a metric space is called open if, for any point x in U, there exists a real number ε > 0 such that any point $y\; \backslash in\; M$ satisfying belongs to U. Equivalently, U is open if every point in U has a neighborhood contained in U.

This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

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Topological space A topology $\backslash tau$ on a set is a set of subsets of with the following properties:

• $X\; \backslash in\; \backslash tau$ and $\backslash varnothing\; \backslash in\; \backslash tau$.
• Any union of sets in $\backslash tau$ belong to $\backslash tau$: if $\backslash left\backslash \{\; U\_i\; :\; i\; \backslash in\; I\; \backslash right\backslash \}\; \backslash subseteq\; \backslash tau$ then $$\backslash bigcup\_\{i\; \backslash in\; I\}\; U\_i\; \backslash in\; \backslash tau\; \backslash ,.$$
• Any finite intersection of sets in $\backslash tau$ belong to $\backslash tau$: for a positive integer $n$, if $U\_1,\; \backslash ldots,\; U\_n\; \backslash in\; \backslash tau$ then $$U\_1\; \backslash cap\; \backslash cdots\; \backslash cap\; U\_n\; \backslash in\; \backslash tau\; \backslash ,.$$

Each member of $\backslash tau$ is called an open set. The set together with $\backslash tau$ is called a topological space.

Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form $\backslash left(\; -1/n,\; 1/n\; \backslash right),$ where $n$ is a positive integer, is the set $\backslash \{\; 0\; \backslash \}$, which is not open in the real line.

A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological spaces that are not metric spaces.

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Properties The union of any number of open sets, or infinitely many open sets, is open.

Joseph L. Taylor (2026). 9780821869017, American Mathematical Society. ISBN 9780821869017

The intersection of a finite number of open sets is open.

A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed.

Steven G. Krantz (2026). 9781420089745, CRC Press. ISBN 9781420089745

A set can never been considered as open by itself. This notion is relative to a containing set and a specific topology on it.

Whether a set is open depends on the topology under consideration. Having opted for greater brevity over greater clarity, we refer to a set X endowed with a topology $\backslash tau$ as "the topological space X" rather than "the topological space $(X,\; \backslash tau)$", despite the fact that all the topological data is contained in $\backslash tau.$ If there are two topologies on the same set, a set U that is open in the first topology might fail to be open in the second topology. For example, if X is any topological space and Y is any subset of X, the set Y can be given its own topology (called the 'subspace topology') defined by "a set U is open in the subspace topology on Y if and only if U is the intersection of Y with an open set from the original topology on X." This potentially introduces new open sets: if V is open in the original topology on X, but $V\; \backslash cap\; Y$ isn't open in the original topology on X, then $V\; \backslash cap\; Y$ is open in the subspace topology on Y.

As a concrete example of this, if U is defined as the set of rational numbers in the interval $(0,\; 1),$ then U is an open subset of the , but not of the real numbers. This is because when the surrounding space is the rational numbers, for every point x in U, there exists a positive number ε such that all points within distance ε of x are also in U. On the other hand, when the surrounding space is the reals, then for every point x in U there is positive ε such that all points within distance ε of x are in U (because U contains no non-rational numbers).

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Uses Open sets have a fundamental importance in topology. The concept is required to define and make sense of topological space and other topological structures that deal with the notions of closeness and convergence for spaces such as metric spaces and uniform spaces.

Every subset A of a topological space X contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A.

A function $f\; :\; X\; \backslash to\; Y$ between two topological spaces $X$ and $Y$ is if the preimage of every open set in $Y$ is open in $X.$ The function $f\; :\; X\; \backslash to\; Y$ is called if the image of every open set in $X$ is open in $Y.$

An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.

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Special types of open sets

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Clopen sets and non-open and/or non-closed sets A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subset a closed subset. Such subsets are known as . Explicitly, a subset $S$ of a topological space $(X,\; \backslash tau)$ is called if both $S$ and its complement $X\; \backslash setminus\; S$ are open subsets of $(X,\; \backslash tau)$; or equivalently, if $S\; \backslash in\; \backslash tau$ and $X\; \backslash setminus\; S\; \backslash in\; \backslash tau.$

In topological space $(X,\; \backslash tau),$ the empty set $\backslash varnothing$ and the set $X$ itself are always clopen. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in topological space. To see, it suffices to remark that, by definition of a topology, $X$ and $\backslash varnothing$ are both open, and that they are also closed, since each is the complement of the other.

The open sets of the usual Euclidean topology of the real line $\backslash R$ are the empty set, the and every union of open intervals.

• The interval $I\; =\; (0,\; 1)$ is open in $\backslash R$ by definition of the Euclidean topology. It is not closed since its complement in $\backslash R$ is $I^\backslash complement\; =\; (-\backslash infty,\; 0]\; \backslash cup\; [1,\; \backslash infty),$ which is not open; indeed, an open interval contained in $I^\backslash complement$ cannot contain , and it follows that $I^\backslash complement$ cannot be a union of open intervals. Hence, $I$ is an example of a set that is open but not closed.
• By a similar argument, the interval $J\; =\; 0,$ is a closed subset but not an open subset.
• Finally, neither $K\; =\; [0,\; 1)$ nor its complement $\backslash R\; \backslash setminus\; K\; =\; (-\backslash infty,\; 0)\; \backslash cup\; [1,\; \backslash infty)$ are open (because they cannot be written as a union of open intervals); this means that $K$ is neither open nor closed.

If a topological space $X$ is endowed with the discrete topology (so that by definition, every subset of $X$ is open) then every subset of $X$ is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that $\backslash mathcal\{U\}$ is an ultrafilter on a non-empty set $X.$ Then the union $\backslash tau\; :=\; \backslash mathcal\{U\}\; \backslash cup\; \backslash \{\; \backslash varnothing\; \backslash \}$ is a topology on $X$ with the property that non-empty proper subset $S$ of $X$ is an open subset or else a closed subset, but never both; that is, if $\backslash varnothing\; \backslash neq\; S\; \backslash subsetneq\; X$ (where $S\; \backslash neq\; X$) then of the following two statements is true: either (1) $S\; \backslash in\; \backslash tau$ or else, (2) $X\; \backslash setminus\; S\; \backslash in\; \backslash tau.$ Said differently, subset is open or closed but the subsets that are both (i.e. that are clopen) are $\backslash varnothing$ and $X.$

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Regular open sets A subset $S$ of a topological space $X$ is called a ' if $\backslash operatorname\{Int\}\; \backslash left(\; \backslash overline\{S\}\; \backslash right)\; =\; S$ or equivalently, if $\backslash operatorname\{Bd\}\; \backslash left(\; \backslash overline\{S\}\; \backslash right)\; =\; \backslash operatorname\{Bd\}\; S$, where $\backslash operatorname\{Bd\}\; S$, $\backslash operatorname\{Int\}\; S$, and $\backslash overline\{S\}$ denote, respectively, the topological boundary, interior, and closure of $S$ in $X$. A topological space for which there exists a base consisting of regular open sets is called a '. A subset of $X$ is a regular open set if and only if its complement in $X$ is a regular closed set, where by definition a subset $S$ of $X$ is called a if $\backslash overline\{\backslash operatorname\{Int\}\; S\}\; =\; S$ or equivalently, if $\backslash operatorname\{Bd\}\; \backslash left(\; \backslash operatorname\{Int\}\; S\; \backslash right)\; =\; \backslash operatorname\{Bd\}\; S.$ Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general,One exception is if $X$ is endowed with the discrete topology, in which case every subset of $X$ is both a regular open subset and a regular closed subset of $X.$ the converses are true.

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Generalizations of open sets Throughout, $(X,\; \backslash tau)$ will be a topological space.

A subset $A\; \backslash subseteq\; X$ of a topological space $X$ is called:

• ' if $A\; ~\backslash subseteq~\; \backslash operatorname\{int\}\_X\; \backslash left(\; \backslash operatorname\{cl\}\_X\; \backslash left(\; \backslash operatorname\{int\}\_X\; A\; \backslash right)\; \backslash right)$, and the complement of such a set is called '.
• ', ', or if it satisfies any of the following equivalent conditions:
  1. $A\; ~\backslash subseteq~\; \backslash operatorname\{int\}\_X\; \backslash left(\; \backslash operatorname\{cl\}\_X\; A\; \backslash right).$
  2. There exists subsets $D,\; U\; \backslash subseteq\; X$ such that $U$ is open in $X,$ $D$ is a dense subset of $X,$ and $A\; =\; U\; \backslash cap\; D.$
  3. There exists an open (in $X$) subset $U\; \backslash subseteq\; X$ such that $A$ is a dense subset of $U.$
  The complement of a preopen set is called .
• ' if $A\; ~\backslash subseteq~\; \backslash operatorname\{int\}\_X\; \backslash left(\; \backslash operatorname\{cl\}\_X\; A\; \backslash right)\; ~\backslash cup~\; \backslash operatorname\{cl\}\_X\; \backslash left(\; \backslash operatorname\{int\}\_X\; A\; \backslash right)$. The complement of a b-open set is called '.
• ' or ' if it satisfies any of the following equivalent conditions:
  1. $A\; ~\backslash subseteq~\; \backslash operatorname\{cl\}\_X\; \backslash left(\; \backslash operatorname\{int\}\_X\; \backslash left(\; \backslash operatorname\{cl\}\_X\; A\; \backslash right)\; \backslash right)$
  2. $\backslash operatorname\{cl\}\_X\; A$ is a regular closed subset of $X.$
  3. There exists a preopen subset $U$ of $X$ such that $U\; \backslash subseteq\; A\; \backslash subseteq\; \backslash operatorname\{cl\}\_X\; U.$
  The complement of a β-open set is called .
• if it satisfies any of the following equivalent conditions:
  1. Whenever a sequence in $X$ converges to some point of $A,$ then that sequence is eventually in $A.$ Explicitly, this means that if $x\_\{\backslash bull\}\; =\; \backslash left(\; x\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ is a sequence in $X$ and if there exists some $a\; \backslash in\; A$ is such that $x\_\{\backslash bull\}\; \backslash to\; x$ in $(X,\; \backslash tau),$ then $x\_\{\backslash bull\}$ is eventually in $A$ (that is, there exists some integer $i$ such that if $j\; \backslash geq\; i,$ then $x\_j\; \backslash in\; A$).
  2. $A$ is equal to its in $X,$ which by definition is the set

     $\backslash begin\{alignat\}\{4\}$

     \operatorname{SeqInt}_X A

     &= \{ a \in A ~:~ \text{ whenever a sequence in } X \text{ converges to } a \text{ in } (X, \tau), \text{ then that sequence is eventually in } A \} \\

     &= \{ a \in A ~:~ \text{ there does NOT exist a sequence in } X \setminus A \text{ that converges in } (X, \tau) \text{ to a point in } A \} \\ \end{alignat}
  The complement of a sequentially open set is called '. A subset $S\; \backslash subseteq\; X$ is sequentially closed in $X$ if and only if $S$ is equal to its ', which by definition is the set $\backslash operatorname\{SeqCl\}\_X\; S$ consisting of all $x\; \backslash in\; X$ for which there exists a sequence in $S$ that converges to $x$ (in $X$).
• ' and is said to have ' if there exists an open subset $U\; \backslash subseteq\; X$ such that $A\; \backslash bigtriangleup\; U$ is a Meager set, where $\backslash bigtriangleup$ denotes the symmetric difference.
  • The subset $A\; \backslash subseteq\; X$ is said to have the Baire property in the restricted sense if for every subset $E$ of $X$ the intersection $A\backslash cap\; E$ has the Baire property relative to $E$.
• if $A\; ~\backslash subseteq~\; \backslash operatorname\{cl\}\_X\; \backslash left(\; \backslash operatorname\{int\}\_X\; A\; \backslash right)$ or, equivalently, $\backslash operatorname\{cl\}\_X\; A\; =\; \backslash operatorname\{cl\}\_X\; \backslash left(\; \backslash operatorname\{int\}\_X\; A\; \backslash right)$. The complement in $X$ of a semi-open set is called a set.
  • The (in $X$) of a subset $A\; \backslash subseteq\; X,$ denoted by $\backslash operatorname\{sCl\}\_X\; A,$ is the intersection of all semi-closed subsets of $X$ that contain $A$ as a subset.
• if for each $x\; \backslash in\; A$ there exists some semiopen subset $U$ of $X$ such that $x\; \backslash in\; U\; \backslash subseteq\; \backslash operatorname\{sCl\}\_X\; U\; \backslash subseteq\; A.$
• ' (resp. ') if its complement in $X$ is a θ-closed (resp. ) set, where by definition, a subset of $X$ is called ' (resp. ') if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point $x\; \backslash in\; X$ is called a ' (resp. a ') of a subset $B\; \backslash subseteq\; X$ if for every open neighborhood $U$ of $x$ in $X,$ the intersection $B\; \backslash cap\; \backslash operatorname\{cl\}\_X\; U$ is not empty (resp. $B\; \backslash cap\; \backslash operatorname\{int\}\_X\backslash left(\; \backslash operatorname\{cl\}\_X\; U\; \backslash right)$ is not empty).

Using the fact that

$A\; ~\backslash subseteq~\; \backslash operatorname\{cl\}\_X\; A\; ~\backslash subseteq~\; \backslash operatorname\{cl\}\_X\; B$ and $\backslash operatorname\{int\}\_X\; A\; ~\backslash subseteq~\; \backslash operatorname\{int\}\_X\; B\; ~\backslash subseteq~\; B$

whenever two subsets $A,\; B\; \backslash subseteq\; X$ satisfy $A\; \backslash subseteq\; B,$ the following may be deduced:

• Every α-open subset is semi-open, semi-preopen, preopen, and b-open.
• Every b-open set is semi-preopen (i.e. β-open).
• Every preopen set is b-open and semi-preopen.
• Every semi-open set is b-open and semi-preopen.

Moreover, a subset is a regular open set if and only if it is preopen and semi-closed. The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set is a semi-preopen (resp. semi-open, preopen, b-open) set. Preopen sets need not be semi-open and semi-open sets need not be preopen.

Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen). However, finite intersections of preopen sets need not be preopen. The set of all α-open subsets of a space $(X,\; \backslash tau)$ forms a topology on $X$ that is finer than $\backslash tau.$

A topological space $X$ is Hausdorff space if and only if every Compact space of $X$ is θ-closed. A space $X$ is totally disconnected if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen. Moreover, the space is totally disconnected if and only if the of every preopen subset is open.

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See also

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Notes

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Bibliography

• Klaas Hart (2026). 9780444503558, Elsevier/North-Holland. ISBN 9780444503558
• (2026). 9780444503558, Elsevier. ISBN 9780444503558

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