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Directed set

In mathematics, a directed set is a nonempty preordered set (I, \leq) in which every finite subset has an upper bound, meaning that for any finite collection of elements in I, there exists an element in I that is greater than or equal to all of them.^[1] Often, the preorder is taken to be a partial order, making the directed set a partially ordered set (poset) where every pair of elements has a common upper bound.^[2] The relation \leq must be reflexive (every element is less than or equal to itself) and transitive (if a \leq b and b \leq c, then a \leq c).^[3] Directed sets generalize the natural numbers under the usual ordering, which serve as index sets for sequences, and extend this concept to more complex structures where "progression" is not necessarily linear.^[4] A dual notion is that of a filtered set (or downward-directed set), where every finite subset has a lower bound instead.^[2] Subsets of a directed set can be cofinal (every element of the original set is bounded above by some element in the subset) or residual (every element bounding the subset from below is in the subset itself), providing tools for analyzing order structures.^[2] Directed sets play a fundamental role in topology, where they index nets—generalizations of sequences that converge in arbitrary topological spaces, enabling characterizations of continuity, compactness, and closure properties beyond metric spaces.^[4] They also appear in order theory for studying completeness (e.g., directed completeness, where every directed subset has a supremum) and in category theory for defining limits and colimits over preordered index categories.^[5] Examples include the set of positive real numbers under the usual order, the collection of finite subsets of a set ordered by inclusion, and the open neighborhoods of a point in a topological space.^[4]

Fundamentals

Definition

A partially ordered set, or poset, is a nonempty set D equipped with a binary relation \leq that is reflexive (a \leq a for all a \in D), antisymmetric (a \leq b and b \leq a imply a = b), and transitive (a \leq b and b \leq c imply a \leq c). A directed set is such a poset in which every pair of elements has an upper bound in D.^[6] Formally, D is directed if for all a, b \in D, there exists c \in D such that a \leq c and b \leq c.^[6] This condition ensures that finite subsets of D also possess upper bounds, as the existence for pairs extends inductively to larger finite collections via the transitivity of \leq.^[4] The standard notion of a directed set is upward-directed, focusing on upper bounds as defined above. A downward-directed set, or dually directed set, is one where every pair of elements has a lower bound c \in D such that c \leq a and c \leq b; this is equivalent to the original set being upward-directed under the reverse order \geq.^[7]

Basic Properties

A fundamental property of a directed set (D, \leq) is that every finite subset has an upper bound in D. This follows directly from the definition, which requires an upper bound for every pair of elements, and extends to larger finite subsets by induction on the cardinality of the subset. For a singleton \{x\}, x itself serves as the upper bound. Assuming the property holds for subsets of size n-1, consider a subset of size n = \{x_1, \dots, x_n\}; let u be an upper bound for \{x_1, \dots, x_{n-1}\}, then there exists v \in D such that u \leq v and x_n \leq v, making v an upper bound for the full subset.^[8] The empty set is vacuously directed, as there are no pairs requiring an upper bound, but directed sets are conventionally required to be nonempty; otherwise, the empty set lacks an element to serve as an upper bound for its empty subset, since any upper bound must belong to the set.^[9] For infinite directed sets, the directedness condition applies solely to finite subsets and does not impose upper bounds—or least upper bounds (suprema)—on infinite subsets. This holds regardless of whether the directed set is countably infinite, like the natural numbers under the usual order (where the entire set lacks an upper bound but every finite subset does), or uncountably infinite.^[4] Directed sets need not be chains, i.e., totally ordered sets, since the upper bound condition allows incomparable elements as long as pairs share a common upper bound. Every chain, however, is a directed set, as the larger of any two elements serves as their upper bound. Under certain conditions, such as those in the prerequisites for Zorn's lemma (where every chain in the poset has an upper bound), directed sets may contain cofinal chains, though this requires the axiom of choice and does not hold generally.^[9]^[10]

Examples

Subset Inclusion

A canonical example of a directed set arises from the power set \mathcal{P}(X) of any set X, equipped with the partial order of subset inclusion, where A \leq B if and only if A \subseteq B.^[11] This ordering makes \mathcal{P}(X) a poset, as inclusion is reflexive, antisymmetric, and transitive.^[11] The poset \mathcal{P}(X) is directed because, for any subsets A, B \subseteq X, their union A \cup B provides an upper bound: A \subseteq A \cup B and B \subseteq A \cup B.^[11] In fact, \mathcal{P}(X) is a complete lattice under this order, with unions as joins and intersections as meets, ensuring upper bounds exist for arbitrary collections as well.^[11] This structure is directed for any set X; however, if X = \emptyset, then \mathcal{P}(X) = \{\emptyset\} is a singleton poset, which is directed in a trivial sense since there are no pairs requiring an upper bound.^[11] For non-empty X, the directedness is non-trivial and illustrates how inclusion captures approximations building toward larger sets. Directed families of subsets extend this idea: a collection \mathcal{F} \subseteq \mathcal{P}(X) is directed if it forms a directed subset of \mathcal{P}(X) under inclusion, meaning any two members of \mathcal{F} have an upper bound within \mathcal{F}.^[11] Such families often model increasing approximations in set-theoretic constructions.

Product of Directed Sets

The product of a family of directed sets \{D_i \mid i \in I\}, where I is an arbitrary index set, is the Cartesian product \prod_{i \in I} D_i equipped with the product order defined by (x_i)_{i \in I} \leq (y_i)_{i \in I} if and only if x_i \leq y_i in D_i for every i \in I.^[12] This defines a partial order on the product set, extending the orders from each component.^[12] Since each D_i is directed, the product \prod_{i \in I} D_i is also directed under this order. To verify directedness, consider any two elements a = (a_i)_{i \in I} and b = (b_i)_{i \in I} in the product. For each i \in I, the directedness of D_i guarantees an upper bound c_i \in D_i such that a_i \leq c_i and b_i \leq c_i. The tuple c = (c_i)_{i \in I} then serves as an upper bound for both a and b in the product, as a_j \leq c_j and b_j \leq c_j hold for all j \in I.^[13] This construction applies to arbitrary index sets I, encompassing both finite and infinite products; finite products form a special case where the verification follows identically by taking componentwise upper bounds.^[13] For instance, the Cartesian product of finite chains—totally ordered directed sets such as \mathbb{N} under the usual order—is directed under the product order, providing a simple example of how directedness propagates. Similarly, if each D_i is the power set of a set under inclusion (a directed set, as covered in the subset inclusion construction), the product consists of families of subsets with the componentwise order, remaining directed.^[13]

Directed Towards a Point

In order theory, a directed set can model convergence towards a limit point p in a set X by considering the poset of subsets containing p, ordered by reverse inclusion. Specifically, let \mathcal{S}_p = \{ A \subseteq X \mid p \in A \} be the collection of all subsets of X containing p. Define the order relation A \leq B if and only if B \subseteq A for A, B \in \mathcal{S}_p. This makes (\mathcal{S}_p, \leq) a partially ordered set, as the relation is reflexive and transitive due to the properties of set inclusion.^[4] This poset is directed because for any finite collection A_1, \dots, A_n \in \mathcal{S}_p, their intersection \bigcap_{i=1}^n A_i also belongs to \mathcal{S}_p (since p is in each A_i) and serves as an upper bound: A_i \leq \bigcap_{j=1}^n A_j for each i, as the intersection is contained in each A_i. In particular, for any pair A, B \in \mathcal{S}_p, A \cap B provides a common upper bound, ensuring the directedness property. This construction generalizes the notion of shrinking sets around p, where upper bounds correspond to refinements that remain centered at p.^[4] A variant employs forward inclusion to model approaching p via increasing approximations. Consider the same collection \mathcal{S}_p, but now ordered by standard inclusion: A \leq B if and only if A \subseteq B. This poset is directed, as the union A \cup B contains p and bounds A and B from above for any pair. Such forward-directed structures arise in filter theory, where the principal filter generated by \{p\} consists of all supersets of \{p\}, facilitating the study of limits through ascending chains or sequences that accumulate at p. This framework motivates the use of directed sets to generalize sequences and filters in convergence: a net indexed by (\mathcal{S}_p, \leq) (in the reverse order) converges to p if it eventually lies in every "small" neighborhood, mirroring the behavior of sequences approaching a limit point in metric spaces or posets.^[14]

Maximal and Greatest Elements

In a partially ordered set (D, \leq), a greatest element g \in D is an element that satisfies d \leq g for all d \in D^[15]. In contrast, a maximal element m \in D is an element such that there is no e \in D with m < e, meaning no strictly larger element exists above it^[15]. These concepts are distinct: every greatest element is maximal, but a maximal element need not be greater than or equal to all others in the set^[15]. Not every directed set possesses a greatest element. For instance, the set of natural numbers \mathbb{N} under the usual order \leq forms a directed set, as any two elements n, m \in \mathbb{N} have an upper bound \max(n, m), but it lacks a greatest element since the order is unbounded above^[16]. However, maximal elements may exist in certain directed sets. By Zorn's lemma, if every chain in the directed set has an upper bound in the set (i.e., the set is chain-complete or inductive), then it contains at least one maximal element^[15]. This condition leverages the directed property, where upper bounds exist for finite subsets like pairs, but extends it to chains for the lemma's application^[15]. Examples illustrate these elements in familiar directed sets. In the power set \mathcal{P}(X) of a set X, ordered by inclusion \subseteq, the set X itself serves as the greatest element, since every subset A \subseteq X satisfies A \subseteq X^[15]. For the product of directed sets, say D_1 \times D_2 under the componentwise order (d_1, d_2) \leq (e_1, e_2) if d_1 \leq_1 e_1 and d_2 \leq_2 e_2, a greatest element exists if and only if each component does; in that case, it is the tuple of the componentwise greatest elements^[17]. If a directed set has a greatest element, it is unique. Suppose g and h are both greatest; then g \leq h and h \leq g, so g = h by antisymmetry of the partial order^[15].

In Logic

In mathematical logic, directed sets serve as foundational structures for modeling approximations in forcing, generic extensions, and denotational semantics of proof systems. In forcing, particularly Cohen forcing, the poset of conditions consists of finite partial functions from \omega to \{0,1\}, partially ordered by reverse inclusion, where a condition p \leq q means p extends q and thus provides a stronger approximation to the generic real being added. Stronger conditions act as upper bounds in this order, refining the information about the eventual generic object, and the collection of conditions compatible with a fixed condition forms a directed set, as any finite compatible subset admits a common extension. This structure models approximations converging toward the generic subset added to the ground model. Generic filters in forcing are maximal directed filters on the poset that intersect every dense subset definable in the ground model, ensuring the extension satisfies the forcing conditions without collapsing cardinals in the case of Cohen forcing. Such generics can be characterized through intersections of directed families of conditions that are dense in the poset, generating the ultrafilter whose union defines the new set in the extension. In proof systems, directed sets model approximations to proofs, where finite partial proofs form a directed poset under extension, with suprema representing complete derivations in denotational semantics.^[18] Extensions refine these approximations, preserving logical validity through continuity in the domain-theoretic interpretation. Historically, directed sets gained prominence in logic through Dana Scott's development of domain theory in the 1970s, providing a lattice-theoretic foundation for denotational semantics of programming languages and recursive function definitions. In Scott's framework, domains are complete partial orders where every directed set has a least upper bound, enabling the fixed-point semantics of recursive proofs and computations as limits of directed chains of approximations. This approach, initiated in Scott's 1970 work on data types as lattices^[19] and formalized in his 1972 paper on continuous lattices,^[20] underpins models for higher-order logic and typed lambda calculi.

Applications

Tails of Nets

In topology, a net generalizes the concept of a sequence by defining a function (x_\alpha)_{\alpha \in D} from a directed set D to a topological space X, where the directed order on D allows for the notion of "eventual" behavior analogous to tails of sequences.^[21] For a net (x_\alpha)_{\alpha \in D}, the tails are the subsets T_\alpha = \{ x_\beta \mid \beta \geq \alpha \} \subseteq X, indexed by elements \alpha \in D. These tails form a partially ordered set under reverse inclusion, where T_\alpha \leq T_\gamma if and only if T_\alpha \supseteq T_\gamma, meaning later tails (with larger indices) are smaller sets contained in earlier ones.^[22] This poset of tails is directed: for any two tails T_\alpha and T_\gamma, there exists an upper bound \delta \in D with \delta \geq \alpha and \delta \geq \gamma, so T_\delta \subseteq T_\alpha and T_\delta \subseteq T_\gamma, hence T_\delta \leq T_\alpha and T_\delta \leq T_\gamma in the reverse inclusion order.^[23] A net (x_\alpha)_{\alpha \in D} converges to a point x \in X if and only if every neighborhood of x contains some tail T_\alpha. This tail-based criterion captures convergence in general topological spaces, where sequences may fail to suffice.^[21]

Neighborhoods in Topology

In a topological space X, the collection \mathcal{N}(x) of all open neighborhoods of a point x \in X forms a partially ordered set under reverse inclusion, where U \leq V if and only if U \supseteq V.^[24] This ordering reflects the refinement of neighborhoods, with finer (smaller) neighborhoods being greater in the poset.^[25] The poset (\mathcal{N}(x), \leq) is directed: for any finite collection of neighborhoods U_1, \dots, U_n \in \mathcal{N}(x), their intersection U_1 \cap \cdots \cap U_n is a nonempty open neighborhood of x and serves as an upper bound, since U_i \leq U_1 \cap \cdots \cap U_n for each i.^[24] This directedness arises from the topological axioms ensuring that finite intersections of open sets remain open and contain x.^[25] A local base at x is a directed subset \mathcal{B}(x) \subseteq \mathcal{N}(x) such that every neighborhood U \in \mathcal{N}(x) contains some B \in \mathcal{B}(x).^[24] Such bases provide a minimal directed system generating the full neighborhood poset, facilitating local descriptions of the topology. This structure underpins the definition of continuity: a function f: X \to Y between topological spaces is continuous at x if the inverse image f^{-1}(V) of every neighborhood V of f(x) contains a neighborhood U of x, preserving the directed refinement of neighborhoods under inverse images.^[25] The neighborhoods of x are directed towards the point x itself, as their intersections approximate the singleton \{x\}. In extensions to locale theory and pointless topology from the 1980s onward, neighborhood systems are generalized to frame elements without reference to points, where directed sets model approximations and completeness in these point-free settings.^[26]

Finite Subsets and Approximations

The poset of all finite subsets of a set X, denoted \mathrm{Fin}(X) and ordered by inclusion, forms a directed set. For any two elements A, B \in \mathrm{Fin}(X), their union A \cup B is finite and acts as an upper bound, as A \subseteq A \cup B and B \subseteq A \cup B. This structure preserves directedness even when restricted to bounded contexts, such as finite unions within a larger poset where finiteness is maintained.^[27] In analysis and algebra, the directed poset \mathrm{Fin}(X) plays a key role in approximations by enabling the formation of inductive limits or colimits in category theory. These limits allow finite subsets to serve as building blocks for larger objects, such as in the completion of metric spaces or the construction of algebraic varieties, where a directed system of finite approximations yields the full structure as a colimit. The rational numbers \mathbb{Q}, equipped with the standard order \leq, exemplify a directed set used for approximations, as any two rationals p, q \in \mathbb{Q} admit an upper bound such as \max(p, q) + 1. Finite increasing sequences of rationals approximate real numbers through mechanisms like supremums in Dedekind cuts, where directed families of such sequences define the cuts that construct \mathbb{R}. This approach highlights how directed sets facilitate dense approximations in real analysis.^[28] Finitely generated directed sets, characterized by a finite generating subset whose upper closure spans the entire poset, connect to free constructions in order theory. Such sets arise as the free join-semilattice on a finite generator set, which is inherently directed and models minimal extensions preserving finite upper bounds. These constructions underpin inductive processes in categorical algebra, linking finite data to broader directed structures.

Directed Subsets

In a partially ordered set (poset) (P, \leq), a subset S \subseteq P is called directed if every finite subset of S has an upper bound in S; that is, for any finite F \subseteq S, there exists s \in S such that f \leq s for all f \in F.^[29] This condition ensures that S, equipped with the order induced from P, forms a directed poset in its own right.^[1] Directed subsets inherit key properties from the broader theory of directed posets, notably the existence of upper bounds for finite collections within the subset itself.^[27] Unlike the ambient poset P, which may not be directed, a directed subset S guarantees that pairwise upper bounds suffice to imply bounds for larger finite sets, as the directedness condition applies recursively.^[2] A directed subset S \subseteq P is cofinal (or unbounded above) if for every p \in P, there exists s \in S such that p \leq s.^[30] Cofinal directed subsets play a crucial role in preserving limits and colimits in order-theoretic constructions, such as when S serves as a "tail" approximating the entire poset.^[31] From a directed subset S, one can generate the upward closure \uparrow S = \{ p \in P \mid \exists s \in S \text{ with } s \leq p \}, which is the smallest upset containing S.^[32] If S is directed, then \uparrow S is an order filter (or simply filter), meaning it is upward closed and every pair of elements in \uparrow S has an upper bound in \uparrow S.^[33] Principal filters generated this way from directed subsets provide a basis for the filter structure on P, facilitating convergence notions in generalized topologies.^[22] Directed subsets relate to ideals in posets through duality: while a directed subset emphasizes upper bounds, an ideal is a downward-directed, downward-closed subset, where every finite subset has a lower bound in the ideal.^[33] This contrast highlights directed subsets as "upward analogs" of ideals, with filters bridging the two via upward closure.^[34] In complete lattices, directed subsets enable the formation of directed unions (suprema of directed families), which are preserved under continuous maps in domain theory and algebraic topology.^[35] For instance, in the context of directed algebraic topology, colimits over directed posets model non-reversible processes, where directed unions of simplicial sets capture homotopy types in categories lacking inverses.^[36] Such constructions ensure compactness in spectral spaces associated with algebraic lattices.^[37]

Contrasts and Relations

With Semilattices

A join-semilattice is a partially ordered set (poset) in which every nonempty finite subset has a least upper bound, known as the join.^[38] In particular, for any two elements x and y, there exists a unique least element x \vee y that is greater than or equal to both, providing a canonical upper bound.^[38] This structure endows the poset with an associative, commutative binary operation corresponding to the join, making it an algebraic semilattice in addition to the order-theoretic one.^[38] Directed sets differ from join-semilattices in that they only require the existence of some upper bound for every finite subset, not necessarily a least one.^[27] Consequently, every join-semilattice is a directed set, as the join serves as an upper bound (and more), but the converse does not hold: there exist directed sets where some pairs lack a least upper bound.^[27] For instance, consider the poset consisting of the sets \{\{a\}, \{d\}, \{a,b,d\}, \{a,c,d\}, \{a,b,c,d\}\} (with distinct elements a,b,c,d), ordered by inclusion.^[39] This is directed, since every pair has a common upper bound (e.g., the full set \{a,b,c,d\}), but the pair \{a\} and \{d\} has no least upper bound, as \{a,b,d\} and \{a,c,d\} are incomparable minimal upper bounds.^[39] Boolean algebras exemplify join-semilattices with rich structure, as they are distributive lattices where every pair has both a join (supremum) and a meet (infimum), complemented by a unary operation.^[40] In contrast, general directed posets, such as the aforementioned set-inclusion poset, lack this uniformity and may not support such operations consistently. This highlights how join-semilattices impose stricter conditions, enabling algebraic manipulations like polynomial identities, whereas directed sets prioritize the weaker property of eventual convergence, sufficient for constructions like nets in analysis without requiring least elements.^[27]

With Filters and Ideals

In order theory, filters on a partially ordered set (poset) (P, \leq) are subsets that extend the notion of directedness with additional closure properties. Specifically, a filter F \subseteq P is a non-empty upward-closed subset that is downward-directed: for any a, b \in F, there exists c \in F such that c \leq a and c \leq b.^[41] This downward-directed condition implies that F, viewed in the opposite poset (P, \geq), is an upward-directed subset, hence a directed set under the reversed order.^[41] Equivalently, in the context of the power set lattice ordered by inclusion, a filter base is an upward-directed collection (every finite subcollection has a common superset in the base) that generates the filter via supersets, with the full filter closed under finite intersections.^[42] Every principal filter, generated by an element x \in P as \uparrow x = \{ y \in P \mid x \leq y \}, is directed: for any finite y_1, \dots, y_n \in \uparrow x, an upper bound exists in \uparrow x if the poset provides one, but in standard settings like the power set, the union serves as such a bound, ensuring directedness.^[41] Ultrafilters represent maximal directed sets in this framework; an ultrafilter on P is a maximal proper filter, meaning no larger proper filter extends it while preserving the closure and directed properties, often corresponding to prime elements in distributive lattices.^[43] Dually, ideals in posets or rings embody downward-directed structures. An ideal I \subseteq P is a non-empty downward-closed subset (if a \in I and b \leq a, then b \in I) that is upward-directed: for any a, b \in I, there exists c \in I such that a \leq c and b \leq c.^[44] In ring theory, ideals like prime ideals satisfy this directedness, with primality ensuring that if the product of ideals lies in the prime ideal, then one factor does, mirroring maximality akin to ultrafilters.^[44] Thus, ideals are directed sets in the original order, providing a downward analogue to the upward focus of filters. While directed sets require only that every finite subset has an upper bound, filters and ideals impose stricter conditions: upward or downward closure, plus closure under finite meets or joins (intersections or unions in set contexts), making them specialized directed subsets for capturing "large" or "small" elements in lattices and rings.^[41] In modern applications, such as the Stone-Čech compactification of a space X, directed filters—particularly ultrafilters—serve as points in the compactification, enabling the universal extension of bounded continuous functions via convergence defined through filter adherence.^[45]

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Let L be a non empty reflexive transitive relational structure. An ideal of L is a directed lower non empty subset of L. A filter of L is a filtered upper non ...
[35]
[PDF] Combinatorics of filters and ideals - UNAM
A family I⊂P(X) of subsets of a given set X is an ideal on X if. (1) for A, B ∈ I, A ∪ B ∈ I,. (2) for A, B ⊂ X, A ⊂ B and B ∈ I implies A ∈ I and. (3) X ∈ I.
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Topology of closure systems in algebraic lattices
Apr 25, 2023 · Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces.
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[PDF] Directed Algebraic Topology - ResearchGate
Directed Algebraic Topology is a recent subject which arose in the. 1990's, on the one hand in abstract settings for homotopy theory, like. [G1], and on the ...
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5 Topology revisited - Paul Taylor
Remark 5.4 Since ΣX carries the Scott topology, a continuous function ΣY→ΣX is a function between open set lattices that preserves directed unions. Such a ...
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filter in nLab
Sep 20, 2025 · Idea. Filters (on a poset) are dual to ideals (in that poset). A proper filter is equivalently the eventuality filter of a net.Definitions · Kinds of filters · Filterbases · The poset of filters and push...
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filter basis - PlanetMath
Mar 22, 2013 · A filter basis B B for a set S S is a non-empty collection of subsets of S S which does not contain the empty set such that, for every u∈B u ...
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ultrafilter in nLab
Aug 18, 2025 · We may also define an ultrafilter to be maximal among the proper filters. ... The composite of these functors gives a monad Set → Set Set \to Set ...
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ideal in nLab
### Summary of Ideal in Order Theory from nLab
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[PDF] pseudotopological spaces and the stone-ˇcech compactification
Many more topological concepts can be expressed directly in terms of (ultra)filters, without explicit reference to open sets. Here is a very important example.