{"history":[],"article":{"id":"6e088477-afe6-47ee-a1a1-2342ca4b21ed","title":"Order topology","slug":"order-topology","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **order [topology](https://bliptext.com/articles/topological-space)** is a specific [topology](https://bliptext.com/articles/topological-space) that can be defined on any [totally ordered set](https://bliptext.com/articles/totally-ordered-set).\nIt is a natural generalization of the topology of the [real numbers](https://bliptext.com/articles/real-number) to arbitrary totally ordered sets.\nIf X is a totally ordered set, the **order topology** on X is generated by the [subbase](https://bliptext.com/articles/subbase) of \"open rays\"\nfor all a, b in X.\nProvided X has at least two elements, this is equivalent to saying that the open [intervals](https://bliptext.com/articles/interval-mathematics)\ntogether with the above rays form a [base](https://bliptext.com/articles/base-topology) for the order topology.\nThe [open sets](https://bliptext.com/articles/open-set) in X are the sets that are a [union](https://bliptext.com/articles/union-set-theory) of (possibly infinitely many) such open intervals and rays.\nA [topological space](https://bliptext.com/articles/topological-space) X is called **orderable** or linearly **orderable** if there exists a total order on its elements such that the order topology induced by that order and the given topology on X coincide.\nThe order topology makes X into a [completely normal](https://bliptext.com/articles/completely-normal-space) [Hausdorff space](https://bliptext.com/articles/hausdorff-space).\nThe standard topologies on **R**, **Q**, **Z**, and **N** are the order topologies.\n\n# Induced order topology\nIf Y is a subset of X, X a totally ordered set, then Y inherits a total order from X.\nThe set Y therefore has an order topology, the **induced order topology**.\nAs a subset of X, Y also has a [subspace topology](https://bliptext.com/articles/subspace-topology).\nThe subspace topology is always at least as [fine](https://bliptext.com/articles/finer-topology) as the induced order topology, but they are not in general the same.\nFor example, consider the subset Y = {−1} ∪ {1/n }n∈N of the [rationals](https://bliptext.com/articles/rational-number).\nUnder the subspace topology, the [singleton set](https://bliptext.com/articles/singleton-set) {−1} is open in Y, but under the induced order topology, any open set containing −1 must contain all but finitely many members of the space.\n\n# Example of a subspace of a linearly ordered space whose topology is not an order topology\nThough the subspace topology of Y = {−1} ∪ {1/n }n∈N in the section above is shown not to be generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the [discrete topology](https://bliptext.com/articles/discrete-topology) on Y (the topology in which every subset of Y is open), and the [discrete topology](https://bliptext.com/articles/discrete-topology) on any set is an order topology.\nTo define a total order on Y that generates the discrete topology on Y, simply modify the induced order on Y by defining −1 to be the greatest element of Y and otherwise keeping the same order for the other points, so that in this new order (call it say <1) we have 1/n <1 −1 for all n ∈ **N**.\nThen, in the order topology on Y generated by <1, every point of Y is isolated in Y.\nWe wish to define here a subset Z of a linearly ordered topological space X such that no total order on Z generates the subspace topology on Z, so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.\nLet in the [real line](https://bliptext.com/articles/real-line).\nThe same argument as before shows that the subspace topology on Z is not equal to the induced order topology on Z, but one can show that the subspace topology on Z cannot be equal to any order topology on Z.\nAn argument follows.\nSuppose by way of contradiction that there is some [strict total order](https://bliptext.com/articles/totally-ordered-set) < on Z such that the order topology generated by < is equal to the subspace topology on Z (note that we are not assuming that < is the induced order on Z, but rather an arbitrarily given total order on Z that generates the subspace topology).\nLet M = Z \\ {−1} = (0,1), then M is [connected](https://bliptext.com/articles/connected-space), so M is dense on itself and has no gaps, in regards to <.\nIf −1 is not the smallest or the largest element of Z, then and (-1,\\infty) separate M, a contradiction.\nAssume without loss of generality that −1 is the smallest element of Z.\nSince {−1} is open in Z, there is some point p in M such that the interval (−1,p) is [empty](https://bliptext.com/articles/empty-set), so p is the minimum of M.\nThen M \\ {p} = (0,p) ∪ (p,1) is not connected with respect to the subspace topology inherited from\nR\n. On the other hand, the subspace topology of M \\ {p} inherited from the order topology of Z coincides with the order topology of M \\ {p} induced by <, which is connected since there are no gaps in M \\ {p} and it is dense.\nThis is a contradiction.\n\n# Left and right order topologies\nSeveral variants of the order topology can be given:\nThe left and right order topologies can be used to give [counterexamples](https://bliptext.com/articles/counterexample) in general topology.\nFor example, the left or right order topology on a bounded set provides an example of a [compact space](https://bliptext.com/articles/compact-space) that is not Hausdorff.\nThe left order topology is the standard topology used for many [set-theoretic](https://bliptext.com/articles/set-theoretic) purposes on a [Boolean algebra](https://bliptext.com/articles/boolean-algebra-structure).\n\n# Ordinal space\nFor any [ordinal number](https://bliptext.com/articles/ordinal-number) λ one can consider the spaces of [ordinal number](https://bliptext.com/articles/ordinal-number)s\ntogether with the natural order topology.\nThese spaces are called **ordinal spaces**.\n(Note that in the usual set-theoretic construction of ordinal numbers we have λ = [0, λ) and λ + 1 = [0, λ]). Obviously, these spaces are mostly of interest when λ is an infinite ordinal; for finite ordinals, the order topology is simply the [discrete topology](https://bliptext.com/articles/discrete-topology).\nWhen λ = ω (the first infinite ordinal), the space [0,ω) is just ****N**** with the usual (still discrete) topology, while [0,ω] is the [one-point compactification](https://bliptext.com/articles/alexandroff-extension) of ****N****.\nOf particular interest is the case when λ = ω1, the set of all countable ordinals, and the [first uncountable ordinal](https://bliptext.com/articles/first-uncountable-ordinal).\nThe element ω1 is a [limit point](https://bliptext.com/articles/limit-point) of the subset [0,ω1) even though no [sequence](https://bliptext.com/articles/sequence) of elements in [0,ω1) has the element ω1 as its limit.\nIn particular, [0,ω1] is not [first-countable](https://bliptext.com/articles/first-countable-space).\nThe subspace [0,ω1) is first-countable however, since the only point in [0,ω1] without a countable [local base](https://bliptext.com/articles/local-base) is ω1. Some further properties include\n\n# Topology and ordinals\n\n## Ordinals as topological spaces\nAny [ordinal number](https://bliptext.com/articles/ordinal-number) can be viewed as a topological space by endowing it with the order topology (indeed, ordinals are [well-ordered](https://bliptext.com/articles/well-order), so in particular [totally ordered](https://bliptext.com/articles/totally-ordered)).\nUnless otherwise specified, this is the usual topology given to ordinals.\nMoreover, if we are willing to accept a [proper class](https://bliptext.com/articles/proper-class) as a topological space, then we may similarly view the class of all ordinals as a topological space with the order topology.\nThe set of [limit points](https://bliptext.com/articles/limit-point) of an ordinal α is precisely the set of [limit ordinals](https://bliptext.com/articles/limit-ordinal) less than α.\n[Successor ordinals](https://bliptext.com/articles/successor-ordinal) (and zero) less than α are [isolated points](https://bliptext.com/articles/isolated-point) in α.\nIn particular, the finite ordinals and ω are [discrete](https://bliptext.com/articles/discrete-space) topological spaces, and no ordinal beyond that is [discrete](https://bliptext.com/articles/discrete-space).\nThe ordinal α is [compact](https://bliptext.com/articles/compact-space) as a topological space if and only if α is either a [successor ordinal](https://bliptext.com/articles/successor-ordinal) or zero.\nThe [closed sets](https://bliptext.com/articles/closed-set) of a limit ordinal α are just the [closed sets](https://bliptext.com/articles/closed-set) in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it.\nAny ordinal is, of course, an open subset of any larger ordinal.\nWe can also define the topology on the ordinals in the following [inductive](https://bliptext.com/articles/recursion) way: 0 is the empty topological space, α+1 is obtained by taking the [one-point compactification](https://bliptext.com/articles/compactification-mathematics) of α, and for δ a limit ordinal, δ is equipped with the [inductive](https://bliptext.com/articles/recursion) limit topology.\nNote that if α is a successor ordinal, then α is compact, in which case its one-point compactification α+1 is the [disjoint union](https://bliptext.com/articles/disjoint-union) of α and a point.\nAs topological spaces, all the ordinals are [Hausdorff](https://bliptext.com/articles/hausdorff-space) and even [normal](https://bliptext.com/articles/normal-space).\nThey are also [totally disconnected](https://bliptext.com/articles/totally-disconnected) (connected components are points), [scattered](https://bliptext.com/articles/scattered-space) (every non-empty subspace has an isolated point; in this case, just take the smallest element), [zero-dimensional](https://bliptext.com/articles/zero-dimensional-space) (the topology has a [clopen](https://bliptext.com/articles/clopen) [basis](https://bliptext.com/articles/basis-topology): here, write an open interval (β,γ) as the union of the [clopen](https://bliptext.com/articles/clopen) intervals (β,γ ' +1) = [ β+1,γ '] for γ ' <γ).\nHowever, they are not [extremally disconnected](https://bliptext.com/articles/extremally-disconnected) in general (there are open sets, for example the even numbers from ω, whose [closure](https://bliptext.com/articles/closure-mathematics) is not open).\nThe topological spaces ω1 and its successor ω1+1 are frequently used as textbook examples of uncountable topological spaces.\nFor example, in the topological space ω1+1, the element ω1 is in the closure of the subset ω1 even though no sequence of elements in ω1 has the element ω1 as its limit: an element in ω1 is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.\nThe space ω1 is [first-countable](https://bliptext.com/articles/first-countable-space) but not [second-countable](https://bliptext.com/articles/second-countable-space), and ω1+1 has neither of these two properties, despite being [compact](https://bliptext.com/articles/compact-space).\nIt is also worthy of note that any [continuous function](https://bliptext.com/articles/continuous-function) from ω1 to **R** (the [real line](https://bliptext.com/articles/real-line)) is eventually constant: so the [Stone–Čech compactification](https://bliptext.com/articles/stone-ech-compactification) of ω1 is ω1+1, just as its one-point compactification (in sharp contrast to ω, whose [Stone–Čech compactification](https://bliptext.com/articles/stone-ech-compactification) is much larger than ω).\n\n## Ordinal-indexed sequences\nIf α is a limit ordinal and X is a set, an α-indexed sequence of elements of X merely means a function from α to X.\nThis concept, a **transfinite [sequence](https://bliptext.com/articles/sequence)** or **ordinal-indexed [sequence](https://bliptext.com/articles/sequence)**, is a generalization of the concept of a [sequence](https://bliptext.com/articles/sequence).\nAn ordinary sequence corresponds to the case α = ω.\nIf X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a [net](https://bliptext.com/articles/net-mathematics), in other words, when given any [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) U of x there is an ordinal β < α such that xι is in U for all ι ≥ β.\nOrdinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1 is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.\nHowever, ordinal-indexed sequences are not powerful enough to replace nets (or [filters](https://bliptext.com/articles/filter-mathematics)) in general: for example, on the [Tychonoff plank](https://bliptext.com/articles/tychonoff-plank) (the product space ), the corner point is a limit point (it is in the closure) of the open subset, but it is not the limit of an ordinal-indexed sequence.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:43:25.445Z","updated_at":"2025-01-25T18:43:25.445Z","search_vector":"'order':1 'topolog':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **order [topology](https://bliptext.com/articles/topological-space)** is a specific [topology](https://bliptext.com/articles/topological-space) that can be defined on any [totally ordered set](https://bliptext.com/articles/totally-ordered-set).\nIt is a natural generalization of the topology of the [real numbers](https://bliptext.com/articles/real-number) to arbitrary totally ordered sets.\nIf X is a totally ordered set, the **order topology** on X is generated by the [subbase](https://bliptext.com/articles/subbase) of \"open rays\"\nfor all a, b in X.\nProvided X has at least two elements, this is equivalent to saying that the open [intervals](https://bliptext.com/articles/interval-mathematics)\ntogether with the above rays form a [base](https://bliptext.com/articles/base-topology) for the order topology.\nThe [open sets](https://bliptext.com/articles/open-set) in X are the sets that are a [union](https://bliptext.com/articles/union-set-theory) of (possibly infinitely many) such open intervals and rays.\nA [topological space](https://bliptext.com/articles/topological-space) X is called **orderable** or linearly **orderable** if there exists a total order on its elements such that the order topology induced by that order and the given topology on X coincide.\nThe order topology makes X into a [completely normal](https://bliptext.com/articles/completely-normal-space) [Hausdorff space](https://bliptext.com/articles/hausdorff-space).\nThe standard topologies on **R**, **Q**, **Z**, and **N** are the order topologies.\n\n# Induced order topology\nIf Y is a subset of X, X a totally ordered set, then Y inherits a total order from X.\nThe set Y therefore has an order topology, the **induced order topology**.\nAs a subset of X, Y also has a [subspace topology](https://bliptext.com/articles/subspace-topology).\nThe subspace topology is always at least as [fine](https://bliptext.com/articles/finer-topology) as the induced order topology, but they are not in general the same.\nFor example, consider the subset Y = {−1} ∪ {1/n }n∈N of the [rationals](https://bliptext.com/articles/rational-number).\nUnder the subspace topology, the [singleton set](https://bliptext.com/articles/singleton-set) {−1} is open in Y, but under the induced order topology, any open set containing −1 must contain all but finitely many members of the space.\n\n# Example of a subspace of a linearly ordered space whose topology is not an order topology\nThough the subspace topology of Y = {−1} ∪ {1/n }n∈N in the section above is shown not to be generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the [discrete topology](https://bliptext.com/articles/discrete-topology) on Y (the topology in which every subset of Y is open), and the [discrete topology](https://bliptext.com/articles/discrete-topology) on any set is an order topology.\nTo define a total order on Y that generates the discrete topology on Y, simply modify the induced order on Y by defining −1 to be the greatest element of Y and otherwise keeping the same order for the other points, so that in this new order (call it say <1) we have 1/n <1 −1 for all n ∈ **N**.\nThen, in the order topology on Y generated by <1, every point of Y is isolated in Y.\nWe wish to define here a subset Z of a linearly ordered topological space X such that no total order on Z generates the subspace topology on Z, so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.\nLet in the [real line](https://bliptext.com/articles/real-line).\nThe same argument as before shows that the subspace topology on Z is not equal to the induced order topology on Z, but one can show that the subspace topology on Z cannot be equal to any order topology on Z.\nAn argument follows.\nSuppose by way of contradiction that there is some [strict total order](https://bliptext.com/articles/totally-ordered-set) < on Z such that the order topology generated by < is equal to the subspace topology on Z (note that we are not assuming that < is the induced order on Z, but rather an arbitrarily given total order on Z that generates the subspace topology).\nLet M = Z \\ {−1} = (0,1), then M is [connected](https://bliptext.com/articles/connected-space), so M is dense on itself and has no gaps, in regards to <.\nIf −1 is not the smallest or the largest element of Z, then and (-1,\\infty) separate M, a contradiction.\nAssume without loss of generality that −1 is the smallest element of Z.\nSince {−1} is open in Z, there is some point p in M such that the interval (−1,p) is [empty](https://bliptext.com/articles/empty-set), so p is the minimum of M.\nThen M \\ {p} = (0,p) ∪ (p,1) is not connected with respect to the subspace topology inherited from\nR\n. On the other hand, the subspace topology of M \\ {p} inherited from the order topology of Z coincides with the order topology of M \\ {p} induced by <, which is connected since there are no gaps in M \\ {p} and it is dense.\nThis is a contradiction.\n\n# Left and right order topologies\nSeveral variants of the order topology can be given:\nThe left and right order topologies can be used to give [counterexamples](https://bliptext.com/articles/counterexample) in general topology.\nFor example, the left or right order topology on a bounded set provides an example of a [compact space](https://bliptext.com/articles/compact-space) that is not Hausdorff.\nThe left order topology is the standard topology used for many [set-theoretic](https://bliptext.com/articles/set-theoretic) purposes on a [Boolean algebra](https://bliptext.com/articles/boolean-algebra-structure).\n\n# Ordinal space\nFor any [ordinal number](https://bliptext.com/articles/ordinal-number) λ one can consider the spaces of [ordinal number](https://bliptext.com/articles/ordinal-number)s\ntogether with the natural order topology.\nThese spaces are called **ordinal spaces**.\n(Note that in the usual set-theoretic construction of ordinal numbers we have λ = [0, λ) and λ + 1 = [0, λ]). Obviously, these spaces are mostly of interest when λ is an infinite ordinal; for finite ordinals, the order topology is simply the [discrete topology](https://bliptext.com/articles/discrete-topology).\nWhen λ = ω (the first infinite ordinal), the space [0,ω) is just ****N**** with the usual (still discrete) topology, while [0,ω] is the [one-point compactification](https://bliptext.com/articles/alexandroff-extension) of ****N****.\nOf particular interest is the case when λ = ω1, the set of all countable ordinals, and the [first uncountable ordinal](https://bliptext.com/articles/first-uncountable-ordinal).\nThe element ω1 is a [limit point](https://bliptext.com/articles/limit-point) of the subset [0,ω1) even though no [sequence](https://bliptext.com/articles/sequence) of elements in [0,ω1) has the element ω1 as its limit.\nIn particular, [0,ω1] is not [first-countable](https://bliptext.com/articles/first-countable-space).\nThe subspace [0,ω1) is first-countable however, since the only point in [0,ω1] without a countable [local base](https://bliptext.com/articles/local-base) is ω1. Some further properties include\n\n# Topology and ordinals\n\n## Ordinals as topological spaces\nAny [ordinal number](https://bliptext.com/articles/ordinal-number) can be viewed as a topological space by endowing it with the order topology (indeed, ordinals are [well-ordered](https://bliptext.com/articles/well-order), so in particular [totally ordered](https://bliptext.com/articles/totally-ordered)).\nUnless otherwise specified, this is the usual topology given to ordinals.\nMoreover, if we are willing to accept a [proper class](https://bliptext.com/articles/proper-class) as a topological space, then we may similarly view the class of all ordinals as a topological space with the order topology.\nThe set of [limit points](https://bliptext.com/articles/limit-point) of an ordinal α is precisely the set of [limit ordinals](https://bliptext.com/articles/limit-ordinal) less than α.\n[Successor ordinals](https://bliptext.com/articles/successor-ordinal) (and zero) less than α are [isolated points](https://bliptext.com/articles/isolated-point) in α.\nIn particular, the finite ordinals and ω are [discrete](https://bliptext.com/articles/discrete-space) topological spaces, and no ordinal beyond that is [discrete](https://bliptext.com/articles/discrete-space).\nThe ordinal α is [compact](https://bliptext.com/articles/compact-space) as a topological space if and only if α is either a [successor ordinal](https://bliptext.com/articles/successor-ordinal) or zero.\nThe [closed sets](https://bliptext.com/articles/closed-set) of a limit ordinal α are just the [closed sets](https://bliptext.com/articles/closed-set) in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it.\nAny ordinal is, of course, an open subset of any larger ordinal.\nWe can also define the topology on the ordinals in the following [inductive](https://bliptext.com/articles/recursion) way: 0 is the empty topological space, α+1 is obtained by taking the [one-point compactification](https://bliptext.com/articles/compactification-mathematics) of α, and for δ a limit ordinal, δ is equipped with the [inductive](https://bliptext.com/articles/recursion) limit topology.\nNote that if α is a successor ordinal, then α is compact, in which case its one-point compactification α+1 is the [disjoint union](https://bliptext.com/articles/disjoint-union) of α and a point.\nAs topological spaces, all the ordinals are [Hausdorff](https://bliptext.com/articles/hausdorff-space) and even [normal](https://bliptext.com/articles/normal-space).\nThey are also [totally disconnected](https://bliptext.com/articles/totally-disconnected) (connected components are points), [scattered](https://bliptext.com/articles/scattered-space) (every non-empty subspace has an isolated point; in this case, just take the smallest element), [zero-dimensional](https://bliptext.com/articles/zero-dimensional-space) (the topology has a [clopen](https://bliptext.com/articles/clopen) [basis](https://bliptext.com/articles/basis-topology): here, write an open interval (β,γ) as the union of the [clopen](https://bliptext.com/articles/clopen) intervals (β,γ ' +1) = [ β+1,γ '] for γ ' <γ).\nHowever, they are not [extremally disconnected](https://bliptext.com/articles/extremally-disconnected) in general (there are open sets, for example the even numbers from ω, whose [closure](https://bliptext.com/articles/closure-mathematics) is not open).\nThe topological spaces ω1 and its successor ω1+1 are frequently used as textbook examples of uncountable topological spaces.\nFor example, in the topological space ω1+1, the element ω1 is in the closure of the subset ω1 even though no sequence of elements in ω1 has the element ω1 as its limit: an element in ω1 is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.\nThe space ω1 is [first-countable](https://bliptext.com/articles/first-countable-space) but not [second-countable](https://bliptext.com/articles/second-countable-space), and ω1+1 has neither of these two properties, despite being [compact](https://bliptext.com/articles/compact-space).\nIt is also worthy of note that any [continuous function](https://bliptext.com/articles/continuous-function) from ω1 to **R** (the [real line](https://bliptext.com/articles/real-line)) is eventually constant: so the [Stone–Čech compactification](https://bliptext.com/articles/stone-ech-compactification) of ω1 is ω1+1, just as its one-point compactification (in sharp contrast to ω, whose [Stone–Čech compactification](https://bliptext.com/articles/stone-ech-compactification) is much larger than ω).\n\n## Ordinal-indexed sequences\nIf α is a limit ordinal and X is a set, an α-indexed sequence of elements of X merely means a function from α to X.\nThis concept, a **transfinite [sequence](https://bliptext.com/articles/sequence)** or **ordinal-indexed [sequence](https://bliptext.com/articles/sequence)**, is a generalization of the concept of a [sequence](https://bliptext.com/articles/sequence).\nAn ordinary sequence corresponds to the case α = ω.\nIf X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a [net](https://bliptext.com/articles/net-mathematics), in other words, when given any [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) U of x there is an ordinal β < α such that xι is in U for all ι ≥ β.\nOrdinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1 is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.\nHowever, ordinal-indexed sequences are not powerful enough to replace nets (or [filters](https://bliptext.com/articles/filter-mathematics)) in general: for example, on the [Tychonoff plank](https://bliptext.com/articles/tychonoff-plank) (the product space ), the corner point is a limit point (it is in the closure) of the open subset, but it is not the limit of an ordinal-indexed sequence.","revisionCount":0},"hasMore":false,"nextCursor":null}

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