{"history":[],"article":{"id":"b6db37c1-ac4c-4445-827e-2c3bba02f5a7","title":"Open set","slug":"open-set","content":":::summary\n:::\n\n# \nred blue circle.svg represents the set of points (x, y) satisfying\nx2 + y2 = r2\n. The red [disk](https://bliptext.com/articles/disk-mathematics) represents the set of points (x, y) satisfying\nx2 + y2 < r2\n. The red set is an open set, the blue set is its [boundary](https://bliptext.com/articles/boundary-topology) set, and the union of the red and blue sets is a [closed set](https://bliptext.com/articles/closed-set).]]\nIn [mathematics](https://bliptext.com/articles/mathematics), an **open set** is a [generalization](https://bliptext.com/articles/generalization) of an [open interval](https://bliptext.com/articles/interval-mathematics) in the [real line](https://bliptext.com/articles/real-line).\nIn a [metric space](https://bliptext.com/articles/metric-space) (a [set](https://bliptext.com/articles/set-mathematics) with a [distance](https://bliptext.com/articles/metric-mathematics) defined between every two points), an open [set](https://bliptext.com/articles/set-mathematics) is a [set](https://bliptext.com/articles/set-mathematics) that, with every point P in it, contains all points of the [metric space](https://bliptext.com/articles/metric-space) that are sufficiently near to P (that is, all points whose [distance](https://bliptext.com/articles/metric-mathematics) to P is less than some value depending on P).\nMore generally, an open set is a member of a given [collection](https://bliptext.com/articles/set-mathematics) of [subsets](https://bliptext.com/articles/subset) of a given set, a [collection](https://bliptext.com/articles/set-mathematics) that has the property of containing every [union](https://bliptext.com/articles/union-set-theory) of its members, every finite [intersection](https://bliptext.com/articles/intersection-set-theory) of its members, the [empty set](https://bliptext.com/articles/empty-set), and the whole set itself.\nA set in which such a collection is given is called a [topological space](https://bliptext.com/articles/topological-space), and the collection is called a [topology](https://bliptext.com/articles/topology-structure).\nThese conditions are very loose, and allow enormous flexibility in the choice of open sets.\nFor example, every subset can be open (the [discrete topology](https://bliptext.com/articles/discrete-topology)), or no subset can be open except the space itself and the empty set (the in[discrete topology](https://bliptext.com/articles/discrete-topology)).\nIn 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.\nIn particular, a topology allows defining properties such as [continuity](https://bliptext.com/articles/continuous-function), [connectedness](https://bliptext.com/articles/connected-space), and [compactness](https://bliptext.com/articles/compactness), which were originally defined by means of a distance.\nThe most common case of a topology without any distance is given by [manifolds](https://bliptext.com/articles/manifold), which are topological spaces that, near each point, resemble an open set of a [Euclidean space](https://bliptext.com/articles/euclidean-space), but on which no distance is defined in general.\nLess intuitive topologies are used in other branches of mathematics; for example, the [Zariski topology](https://bliptext.com/articles/zariski-topology), which is fundamental in [algebraic geometry](https://bliptext.com/articles/algebraic-geometry) and [scheme theory](https://bliptext.com/articles/scheme-theory).\n\n# Motivation\nIntuitively, an open set provides a method to distinguish two [points](https://bliptext.com/articles/point-geometry).\nFor example, if about one of two points in a [topological space](https://bliptext.com/articles/topological-space), there exists an open set not containing the other (distinct) point, the two points are referred to as [topologically distinguishable](https://bliptext.com/articles/topologically-distinguishable).\nIn this manner, one may speak of whether two points, or more generally two [subsets](https://bliptext.com/articles/subset), of a topological space are \"near\" without concretely defining a [distance](https://bliptext.com/articles/metric-mathematics).\nTherefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are called [metric spaces](https://bliptext.com/articles/metric-space).\nIn the set of all [real numbers](https://bliptext.com/articles/real-number), one has the natural [Euclidean metric](https://bliptext.com/articles/euclidean-metric); that is, a function which measures the distance between two [real numbers](https://bliptext.com/articles/real-number):\nd(x, y) =\n. 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.\nIn essence, points within ε of x approximate x to an accuracy of degree ε.\nNote 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.\nFor example, if x = 0 and ε = 1, the points within ε of x are precisely the points of the [interval](https://bliptext.com/articles/interval-mathematics) (−1, 1); that is, the set of all real numbers between −1 and 1.\nHowever, with ε = 0.5, the points within ε of x are precisely the points of (−0.5, 0.5).\nClearly, these points approximate x to a greater degree of accuracy than when ε = 1.\nThe 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.\nIn particular, sets of the form (−ε, ε) give us a lot of information about points close to x = 0.\nThus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x.\nThis 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.\nFor example, if we were to define ****R**** as the only such set for \"measuring distance\", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of ****R****.\nThus, we find that in some sense, every real number is distance 0 away from 0.\nIt may help in this case to think of the measure as being a binary condition: all things in ****R**** are equally close to 0, while any item that is not in ****R**** is not close to 0.\nIn 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.\nIn fact, one may generalize these notions to an arbitrary set (X); rather than just the real numbers.\nIn 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.\nOf 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.\nFor example, every point in X should approximate x to some degree of accuracy.\nThus X should be in this family.\nOnce we begin to define \"smaller\" sets containing x, we tend to approximate x to a greater degree of accuracy.\nBearing this in mind, one may define the remaining axioms that the family of sets about x is required to satisfy.\n\n# Definitions\nSeveral definitions are given here, in an increasing order of technicality.\nEach one is a special case of the next one.\n\n## Euclidean space\nA subset U of the [[Euclidean space|Euclidean\nn\n-space]]\nRn\nis open if, for every point x in U, [there exists](https://bliptext.com/articles/there-exists) a positive real number ε (depending on x) such that any point in\nRn\nwhose [Euclidean distance](https://bliptext.com/articles/euclidean-distance) from x is smaller than ε belongs to U. Equivalently, a subset U of\nRn\nis open if every point in U is the center of an [open ball](https://bliptext.com/articles/open-ball) contained in U.\nAn example of a subset of\nR\nthat is not open is the [closed interval](https://bliptext.com/articles/interval-mathematics), since neither\n0 - ε\nnor\n1 + ε\nbelongs to for any\nε > 0\n, no matter how small.\n\n## Metric space\nA subset U of a [metric space](https://bliptext.com/articles/metric-space)\n(M, d)\nis called open if, for any point x in U, there exists a real number ε > 0 such that any point y \\in M satisfying\nd(x, y) < ε\nbelongs to U.\nEquivalently, U is open if every point in U has a neighborhood contained in U.\nThis generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.\n\n## Topological space\nA topology \\tau on a set X is a set of subsets of X with the properties below.\nEach member of \\tau is called an open set.\nX together with \\tau is called a [topological space](https://bliptext.com/articles/topological-space).\nInfinite intersections of open sets need not be open.\nFor example, the intersection of all intervals of the form where n is a positive integer, is the set \\{ 0 \\} which is not open in the real line.\nA metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls.\nThere are, however, topological spaces that are not metric spaces.\n\n# Properties\nThe [union](https://bliptext.com/articles/union-set-theory) of any number of open sets, or infinitely many open sets, is open.\nThe [intersection](https://bliptext.com/articles/intersection-set-theory) of a finite number of open sets is open.\nA [complement](https://bliptext.com/articles/complement-set-theory) of an open set (relative to the space that the topology is defined on) is called a [closed set](https://bliptext.com/articles/closed-set).\nA set may be both open and closed (a [clopen set](https://bliptext.com/articles/clopen-set)).\nThe [empty set](https://bliptext.com/articles/empty-set) and the full space are examples of sets that are both open and closed.\nA set can never been considered as open by itself.\nThis notion is relative to a containing set and a specific topology on it.\nWhether a set is open depends on the [topology](https://bliptext.com/articles/topology) under consideration.\nHaving opted for [greater brevity over greater clarity](https://bliptext.com/articles/abuse-of-notation), we refer to a set X endowed with a topology \\tau as \"the topological space X\" rather than \"the topological space (X, \\tau)\", despite the fact that all the topological data is contained in \\tau.\nIf 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.\nFor 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.\"\nThis potentially introduces new open sets: if V is open in the original topology on X, but V \\cap Y isn't open in the original topology on X, then V \\cap Y is open in the subspace topology on Y.\nAs a concrete example of this, if U is defined as the set of [rational numbers](https://bliptext.com/articles/rational-number) in the interval (0, 1), then U is an open subset of the [rational numbers](https://bliptext.com/articles/rational-number), but not of the [real numbers](https://bliptext.com/articles/real-numbers).\nThis is because when the surrounding space is the rational numbers, for every point x in U, there exists a positive number a such that all points within distance a of x are also in U.\nOn the other hand, when the surrounding space is the reals, then for every point x in U there is positive a such that all points within distance a of x are in U (because U contains no non-rational numbers).\n\n# Uses\nOpen sets have a fundamental importance in [topology](https://bliptext.com/articles/topology).\nThe concept is required to define and make sense of [topological space](https://bliptext.com/articles/topological-space) and other topological structures that deal with the notions of closeness and convergence for spaces such as [metric spaces](https://bliptext.com/articles/metric-spaces) and [uniform spaces](https://bliptext.com/articles/uniform-spaces).\nEvery [subset](https://bliptext.com/articles/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](https://bliptext.com/articles/topological-interior) of A.\nIt can be constructed by taking the union of all the open sets contained in A.\nA [function](https://bliptext.com/articles/function-mathematics) f : X \\to Y between two topological spaces X and Y is if the [preimage](https://bliptext.com/articles/pre[image](https://bliptext.com/articles/image-mathematics)) of every open set in Y is open in X. The [function](https://bliptext.com/articles/function-mathematics) f : X \\to Y is called if the [image](https://bliptext.com/articles/image-mathematics) of every open set in X is open in Y.\nAn open set on the [real line](https://bliptext.com/articles/real-line) has the characteristic property that it is a countable union of disjoint open intervals.\n\n# Special types of open sets\n\n## Clopen sets and non-open and/or non-closed sets\nA set might be open, closed, both, or neither.\nIn 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.\n****S****u****c****h**** ****s****u****b****s****e****t****s**** ****a****r****e**** ****k****n****o****w****n**** ****a****s**** ****.****\nExplicitly, a subset S of a topological space (X, \\tau) is called if both S and its complement are open subsets of (X, \\tau); or equivalently, if S \\in \\tau and\nIn topological space (X, \\tau), the empty set \\varnothing and the set X itself are always clopen.\nThese two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in topological space.\nTo see, it suffices to remark that, by definition of a topology, X and \\varnothing are both open, and that they are also closed, since each is the complement of the other.\nThe open sets of the usual [Euclidean topology](https://bliptext.com/articles/euclidean-topology) of the [real line](https://bliptext.com/articles/real-line) \\R are the empty set, the [open intervals](https://bliptext.com/articles/open-interval) and every union of [open intervals](https://bliptext.com/articles/open-interval).\n1\n, and it follows that cannot be a union of open intervals.\nHence, I is an example of a set that is open but not closed.\nIf a topological space X is endowed with the [discrete topology](https://bliptext.com/articles/discrete-topology) (so that by definition, every subset of X is open) then every subset of X is a clopen subset.\nFor a more advanced example reminiscent of the discrete topology, suppose that \\mathcal{U} is an [ultrafilter](https://bliptext.com/articles/ultrafilter) on a non-empty set X. Then the union 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 (where S \\neq X) then of the following two statements is true: either (1) S \\in \\tau or else, (2) Said differently, subset is open or closed but the subsets that are both (i.e. that are clopen) are \\varnothing and X.\n\n## Regular open sets\nA subset S of a topological space X is called a **' if or equivalently, if, where , , and denote, respectively, the topological [boundary](https://bliptext.com/articles/boundary-topology), [interior](https://bliptext.com/articles/interior-topology), and [closure](https://bliptext.com/articles/closure-topology) of S in X. A topological space for which there exists a [base](https://bliptext.com/articles/base-topology) consisting of regular open sets is called a **'.\n****A**** ****s****u****b****s****e****t**** ****o****f**** ****X**** ****i****s**** ****a**** ****r****e****g****u****l****a****r**** ****o****p****e****n**** ****s****e****t**** ****i****f**** ****a****n****d**** ****o****n****l****y**** ****i****f**** ****i****t****s**** ****c****o****m****p****l****e****m****e****n****t**** ****i****n**** ****X**** ****i****s**** ****a**** ****r****e****g****u****l****a****r**** ****c****l****o****s****e****d**** ****s****e****t****,**** ****w****h****e****r****e**** ****b****y**** ****d****e****f****i****n****i****t****i****o****n**** ****a**** ****s****u****b****s****e****t**** ****S**** ****o****f**** ****X**** ****i****s**** ****c****a****l****l****e****d**** ****a**** **** ****i****f**** ****o****r**** ****e****q****u****i****v****a****l****e****n****t****l****y****,**** ****i****f**** **** ****E****v****e****r****y**** ****r****e****g****u****l****a****r**** ****o****p****e****n**** ****s****e****t**** ****(****r****e****s****p****.**** ****r****e****g****u****l****a****r**** ****c****l****o****s****e****d**** ****s****e****t****)**** ****i****s**** ****a****n**** ****o****p****e****n**** ****s****u****b****s****e****t**** ****(****r****e****s****p****.**** ****i****s**** ****a**** ****c****l****o****s****e****d**** ****s****u****b****s****e****t****)**** ****a****l****t****h****o****u****g****h**** ****i****n**** ****g****e****n****e****r****a****l****,**** ****t****h****e**** ****c****o****n****v****e****r****s****e****s**** ****a****r****e**** **** ****t****r****u****e****.****\n\n# Generalizations of open sets\nThroughout, (X, \\tau) will be a topological space.\nA subset of a topological space X is called:\n

**' if, and the complement of such a set is called **'.

There exists subsets such that U is open in X, D is a [dense subset](https://bliptext.com/articles/dense-subset) of X, and
There exists an open (in X) subset such that A is a dense subset of U.

**' or **' if it satisfies any of the following equivalent conditions:\n
2. is a regular closed subset of X.
3. There exists a preopen subset U of X such that
\n****T****h****e**** ****c****o****m****p****l****e****m****e****n****t**** ****o****f**** ****a**** ****β****-****o****p****e****n**** ****s****e****t**** ****i****s**** ****c****a****l****l****e****d**** ****.****\n

Whenever a sequence in X converges to some point of A, then that sequence is eventually in A. Explicitly, this means that if is a sequence in X and if there exists some a \\in A is such that in (X, \\tau), then x_{\\bull} is eventually in A (that is, there exists some integer i such that if j \\geq i, then x_j \\in A).

**' and is said to have **' if there exists an open subset such that is a [meager subset](https://bliptext.com/articles/meager-set), where denotes the [symmetric difference](https://bliptext.com/articles/symmetric-difference).\n****<****l****i****>**** ****i****f**** ****o****r****,**** ****e****q****u****i****v****a****l****e****n****t****l****y****,****.****\nThe complement in X of a semi-open** set** is called a ** set**.\n****<****l****i****>**** ****i****f**** ****f****o****r**** ****e****a****c****h**** ****x**** ****\\****i****n**** ****A**** ****t****h****e****r****e**** ****e****x****i****s****t****s**** ****s****o****m****e**** ****s****e****m****i****o****p****e****n**** ****s****u****b****s****e****t**** ****U**** ****o****f**** ****X**** ****s****u****c****h**** ****t****h****a****t**** ****<****/****l****i****>****\n
****' (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).\nA point x \\in X is called a **' (resp. a **') of a subset if for every open neighborhood U of x in X, the intersection is not empty (resp. is not empty).

\nUsing the fact that\nwhenever two subsets satisfy the following may be deduced:\nMoreover, a subset is a regular open set if and only if it is preopen and semi-closed.\nThe 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.\nPreopen sets need not be semi-open and semi-open sets need not be preopen.\nArbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen).\nHowever, finite intersections of preopen sets need not be preopen.\nThe set of all α-open subsets of a space (X, \\tau) forms a topology on X that is [finer](https://bliptext.com/articles/comparison-of-topologies) than \\tau.\nA topological space X is [Hausdorff](https://bliptext.com/articles/hausdorff-space) if and only if every [compact subspace](https://bliptext.com/articles/compact-space) of X is θ-closed.\nA space X is [totally disconnected](https://bliptext.com/articles/totally-disconnected) if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen.\n****M****o****r****e****o****v****e****r****,**** ****t****h****e**** ****s****p****a****c****e**** ****i****s**** ****t****o****t****a****l****l****y**** ****d****i****s****c****o****n****n****e****c****t****e****d**** ****i****f**** ****a****n****d**** ****o****n****l****y**** ****i****f**** ****t****h****e**** **** ****o****f**** ****e****v****e****r****y**** ****p****r****e****o****p****e****n**** ****s****u****b****s****e****t**** ****i****s**** ****o****p****e****n****.****","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:43:15.661Z","updated_at":"2025-01-25T16:43:15.661Z","search_vector":"'open':1 'set':2","search_content":" \n \n\n# \nred blue circle.svg represents the set of points (x, y) satisfying\nx2 + y2 = r2\n. The red [disk](https://bliptext.com/articles/disk-mathematics) represents the set of points (x, y) satisfying\nx2 + y2 < r2\n. The red set is an open set, the blue set is its [boundary](https://bliptext.com/articles/boundary-topology) set, and the union of the red and blue sets is a [closed set](https://bliptext.com/articles/closed-set).]]\nIn [mathematics](https://bliptext.com/articles/mathematics), an **open set** is a [generalization](https://bliptext.com/articles/generalization) of an [open interval](https://bliptext.com/articles/interval-mathematics) in the [real line](https://bliptext.com/articles/real-line).\nIn a [metric space](https://bliptext.com/articles/metric-space) (a [set](https://bliptext.com/articles/set-mathematics) with a [distance](https://bliptext.com/articles/metric-mathematics) defined between every two points), an open [set](https://bliptext.com/articles/set-mathematics) is a [set](https://bliptext.com/articles/set-mathematics) that, with every point P in it, contains all points of the [metric space](https://bliptext.com/articles/metric-space) that are sufficiently near to P (that is, all points whose [distance](https://bliptext.com/articles/metric-mathematics) to P is less than some value depending on P).\nMore generally, an open set is a member of a given [collection](https://bliptext.com/articles/set-mathematics) of [subsets](https://bliptext.com/articles/subset) of a given set, a [collection](https://bliptext.com/articles/set-mathematics) that has the property of containing every [union](https://bliptext.com/articles/union-set-theory) of its members, every finite [intersection](https://bliptext.com/articles/intersection-set-theory) of its members, the [empty set](https://bliptext.com/articles/empty-set), and the whole set itself.\nA set in which such a collection is given is called a [topological space](https://bliptext.com/articles/topological-space), and the collection is called a [topology](https://bliptext.com/articles/topology-structure).\nThese conditions are very loose, and allow enormous flexibility in the choice of open sets.\nFor example, every subset can be open (the [discrete topology](https://bliptext.com/articles/discrete-topology)), or no subset can be open except the space itself and the empty set (the in[discrete topology](https://bliptext.com/articles/discrete-topology)).\nIn 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.\nIn particular, a topology allows defining properties such as [continuity](https://bliptext.com/articles/continuous-function), [connectedness](https://bliptext.com/articles/connected-space), and [compactness](https://bliptext.com/articles/compactness), which were originally defined by means of a distance.\nThe most common case of a topology without any distance is given by [manifolds](https://bliptext.com/articles/manifold), which are topological spaces that, near each point, resemble an open set of a [Euclidean space](https://bliptext.com/articles/euclidean-space), but on which no distance is defined in general.\nLess intuitive topologies are used in other branches of mathematics; for example, the [Zariski topology](https://bliptext.com/articles/zariski-topology), which is fundamental in [algebraic geometry](https://bliptext.com/articles/algebraic-geometry) and [scheme theory](https://bliptext.com/articles/scheme-theory).\n\n# Motivation\nIntuitively, an open set provides a method to distinguish two [points](https://bliptext.com/articles/point-geometry).\nFor example, if about one of two points in a [topological space](https://bliptext.com/articles/topological-space), there exists an open set not containing the other (distinct) point, the two points are referred to as [topologically distinguishable](https://bliptext.com/articles/topologically-distinguishable).\nIn this manner, one may speak of whether two points, or more generally two [subsets](https://bliptext.com/articles/subset), of a topological space are \"near\" without concretely defining a [distance](https://bliptext.com/articles/metric-mathematics).\nTherefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are called [metric spaces](https://bliptext.com/articles/metric-space).\nIn the set of all [real numbers](https://bliptext.com/articles/real-number), one has the natural [Euclidean metric](https://bliptext.com/articles/euclidean-metric); that is, a function which measures the distance between two [real numbers](https://bliptext.com/articles/real-number):\nd(x, y) =\n. 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.\nIn essence, points within ε of x approximate x to an accuracy of degree ε.\nNote 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.\nFor example, if x = 0 and ε = 1, the points within ε of x are precisely the points of the [interval](https://bliptext.com/articles/interval-mathematics) (−1, 1); that is, the set of all real numbers between −1 and 1.\nHowever, with ε = 0.5, the points within ε of x are precisely the points of (−0.5, 0.5).\nClearly, these points approximate x to a greater degree of accuracy than when ε = 1.\nThe 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.\nIn particular, sets of the form (−ε, ε) give us a lot of information about points close to x = 0.\nThus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x.\nThis 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.\nFor example, if we were to define ****R**** as the only such set for \"measuring distance\", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of ****R****.\nThus, we find that in some sense, every real number is distance 0 away from 0.\nIt may help in this case to think of the measure as being a binary condition: all things in ****R**** are equally close to 0, while any item that is not in ****R**** is not close to 0.\nIn 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.\nIn fact, one may generalize these notions to an arbitrary set (X); rather than just the real numbers.\nIn 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.\nOf 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.\nFor example, every point in X should approximate x to some degree of accuracy.\nThus X should be in this family.\nOnce we begin to define \"smaller\" sets containing x, we tend to approximate x to a greater degree of accuracy.\nBearing this in mind, one may define the remaining axioms that the family of sets about x is required to satisfy.\n\n# Definitions\nSeveral definitions are given here, in an increasing order of technicality.\nEach one is a special case of the next one.\n\n## Euclidean space\nA subset U of the [[Euclidean space|Euclidean\nn\n-space]]\nRn\nis open if, for every point x in U, [there exists](https://bliptext.com/articles/there-exists) a positive real number ε (depending on x) such that any point in\nRn\nwhose [Euclidean distance](https://bliptext.com/articles/euclidean-distance) from x is smaller than ε belongs to U. Equivalently, a subset U of\nRn\nis open if every point in U is the center of an [open ball](https://bliptext.com/articles/open-ball) contained in U.\nAn example of a subset of\nR\nthat is not open is the [closed interval](https://bliptext.com/articles/interval-mathematics), since neither\n0 - ε\nnor\n1 + ε\nbelongs to for any\nε > 0\n, no matter how small.\n\n## Metric space\nA subset U of a [metric space](https://bliptext.com/articles/metric-space)\n(M, d)\nis called open if, for any point x in U, there exists a real number ε > 0 such that any point y \\in M satisfying\nd(x, y) \n **' if, and the complement of such a set is called **'. \n**** **************'****,**** ************************'****,**** **************o****r**** **** ****i****f**** ****i****t**** ****s****a****t****i****s****f****i****e****s**** ****a****n****y**** ****o****f**** ****t****h****e**** ****f****o****l****l****o****w****i****n****g**** ****e****q****u****i****v****a****l****e****n****t**** ****c****o****n****d****i****t****i****o****n****s****:****\n \n \n There exists subsets such that U is open in X, D is a [dense subset](https://bliptext.com/articles/dense-subset) of X, and \n There exists an open (in X) subset such that A is a dense subset of U. \n \n****T****h****e**** ****c****o****m****p****l****e****m****e****n****t**** ****o****f**** ****a**** ****p****r****e****o****p****e****n**** ****s****e****t**** ****i****s**** ****c****a****l****l****e****d**** ****.****\n \n**** **** ****i****f****.****\n****T****h****e**** ****c****o****m****p****l****e****m****e****n****t**** ****o****f**** ****a**** ****b****-****o****p****e****n**** ****s****e****t**** ****i****s**** ****c****a****l****l****e****d**** ****.**** ****\n **' or **' if it satisfies any of the following equivalent conditions:\n \n \n is a regular closed subset of X. \n There exists a preopen subset U of X such that \n \n****T****h****e**** ****c****o****m****p****l****e****m****e****n****t**** ****o****f**** ****a**** ****β****-****o****p****e****n**** ****s****e****t**** ****i****s**** ****c****a****l****l****e****d**** ****.****\n \n**** **** ****i****f**** ****i****t**** ****s****a****t****i****s****f****i****e****s**** ****a****n****y**** ****o****f**** ****t****h****e**** ****f****o****l****l****o****w****i****n****g**** ****e****q****u****i****v****a****l****e****n****t**** ****c****o****n****d****i****t****i****o****n****s****:****\n \n Whenever a sequence in X converges to some point of A, then that sequence is eventually in A. Explicitly, this means that if is a sequence in X and if there exists some a \\in A is such that in (X, \\tau), then x_{\\bull} is eventually in A (that is, there exists some integer i such that if j \\geq i, then x_j \\in A). \n**** ****A**** ****i****s**** ****e****q****u****a****l**** ****t****o**** ****i****t****s**** **** ****i****n**** ****X****,**** ****w****h****i****c****h**** ****b****y**** ****d****e****f****i****n****i****t****i****o****n**** ****i****s**** ****t****h****e**** ****s****e****t****\n \n \n****T****h****e**** ****c****o****m****p****l****e****m****e****n****t**** ****o****f**** ****a**** ****s****e****q****u****e****n****t****i****a****l****l****y**** ****o****p****e****n**** ****s****e****t**** ****i****s**** ****c****a****l****l****e****d**** ****.****\n****A**** ****s****u****b****s****e****t**** ****i****s**** ****s****e****q****u****e****n****t****i****a****l****l****y**** ****c****l****o****s****e****d**** ****i****n**** ****X**** ****i****f**** ****a****n****d**** ****o****n****l****y**** ****i****f**** ****S**** ****i****s**** ****e****q****u****a****l**** ****t****o**** ****i****t****s**** ****,**** ****w****h****i****c****h**** ****b****y**** ****d****e****f****i****n****i****t****i****o****n**** ****i****s**** ****t****h****e**** ****s****e****t**** **** ****c****o****n****s****i****s****t****i****n****g**** ****o****f**** ****a****l****l**** ****x**** ****\\****i****n**** ****X**** ****f****o****r**** ****w****h****i****c****h**** ****t****h****e****r****e**** ****e****x****i****s****t****s**** ****a**** ****s****e****q****u****e****n****c****e**** ****i****n**** ****S**** ****t****h****a****t**** ****c****o****n****v****e****r****g****e****s**** ****t****o**** ****x**** ****(****i****n**** ****X****)****.****\n \n **' and is said to have **' if there exists an open subset such that is a [meager subset](https://bliptext.com/articles/meager-set), where denotes the [symmetric difference](https://bliptext.com/articles/symmetric-difference).\n**** **** ****i****f**** ****o****r****,**** ****e****q****u****i****v****a****l****e****n****t****l****y****,****.****\nThe complement in X of a semi-open** set** is called a ** set**.\n**** **** ****i****f**** ****f****o****r**** ****e****a****c****h**** ****x**** ****\\****i****n**** ****A**** ****t****h****e****r****e**** ****e****x****i****s****t****s**** ****s****o****m****e**** ****s****e****m****i****o****p****e****n**** ****s****u****b****s****e****t**** ****U**** ****o****f**** ****X**** ****s****u****c****h**** ****t****h****a****t**** **** ****\n ****' (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).\nA point x \\in X is called a **' (resp. a **') of a subset if for every open neighborhood U of x in X, the intersection is not empty (resp. is not empty). \n \nUsing the fact that\nwhenever two subsets satisfy the following may be deduced:\nMoreover, a subset is a regular open set if and only if it is preopen and semi-closed.\nThe 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.\nPreopen sets need not be semi-open and semi-open sets need not be preopen.\nArbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen).\nHowever, finite intersections of preopen sets need not be preopen.\nThe set of all α-open subsets of a space (X, \\tau) forms a topology on X that is [finer](https://bliptext.com/articles/comparison-of-topologies) than \\tau.\nA topological space X is [Hausdorff](https://bliptext.com/articles/hausdorff-space) if and only if every [compact subspace](https://bliptext.com/articles/compact-space) of X is θ-closed.\nA space X is [totally disconnected](https://bliptext.com/articles/totally-disconnected) if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen.\n****M****o****r****e****o****v****e****r****,**** ****t****h****e**** ****s****p****a****c****e**** ****i****s**** ****t****o****t****a****l****l****y**** ****d****i****s****c****o****n****n****e****c****t****e****d**** ****i****f**** ****a****n****d**** ****o****n****l****y**** ****i****f**** ****t****h****e**** **** ****o****f**** ****e****v****e****r****y**** ****p****r****e****o****p****e****n**** ****s****u****b****s****e****t**** ****i****s**** ****o****p****e****n****.****","revisionCount":0},"hasMore":false,"nextCursor":null}

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