Hubbry Logo
Comparison of topologiesComparison of topologiesMain
Open search
Comparison of topologies
Community hub
Comparison of topologies
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Comparison of topologies
Comparison of topologies
Comparison of topologies
View on Wikipedia
from Wikipedia

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.

Definition

[edit]

A topology on a set may be defined as the collection of subsets which are considered to be "open". (An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.)

For definiteness the reader should think of a topology as the family of open sets of a topological space, since that is the standard meaning of the word "topology".

Let τ1 and τ2 be two topologies on a set X such that τ1 is contained in τ2:

.

That is, every element of τ1 is also an element of τ2. Then the topology τ1 is said to be a coarser (weaker or smaller) topology than τ2, and τ2 is said to be a finer (stronger or larger) topology than τ1. [nb 1]

If additionally

we say τ1 is strictly coarser than τ2 and τ2 is strictly finer than τ1.[1]

The binary relation ⊆ defines a partial ordering relation on the set of all possible topologies on X.

Examples

[edit]

The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set and the whole space as open sets.

In function spaces and spaces of measures there are often a number of possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships.

All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.

The complex vector space Cn may be equipped with either its usual (Euclidean) topology, or its Zariski topology. In the latter, a subset V of Cn is closed if and only if it consists of all solutions to some system of polynomial equations. Since any such V also is a closed set in the ordinary sense, but not vice versa, the Zariski topology is strictly weaker than the ordinary one.

Properties

[edit]

Let τ1 and τ2 be two topologies on a set X. Then the following statements are equivalent:

(The identity map idX is surjective and therefore it is strongly open if and only if it is relatively open.)

Two immediate corollaries of the above equivalent statements are

  • A continuous map f : X → Y remains continuous if the topology on Y becomes coarser or the topology on X finer.
  • An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser.

One can also compare topologies using neighborhood bases. Let τ1 and τ2 be two topologies on a set X and let Bi(x) be a local base for the topology τi at x ∈ X for i = 1,2. Then τ1 ⊆ τ2 if and only if for all x ∈ X, each open set U1 in B1(x) contains some open set U2 in B2(x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.

Lattice of topologies

[edit]

The set of all topologies on a set X together with the partial ordering relation ⊆ forms a complete lattice that is also closed under arbitrary intersections.[2] That is, any collection of topologies on X have a meet (or infimum) and a join (or supremum). The meet of a collection of topologies is the intersection of those topologies. The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.

Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.

The lattice of topologies on a set is a complemented lattice; that is, given a topology on there exists a topology on such that the intersection is the trivial topology and the topology generated by the union is the discrete topology.[3][4]

If the set has at least three elements, the lattice of topologies on is not modular,[5] and hence not distributive either.

See also

[edit]
  • Initial topology, the coarsest topology on a set to make a family of mappings from that set continuous
  • Final topology, the finest topology on a set to make a family of mappings into that set continuous

Notes

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Comparison of topologies

View on Grokipedia
from Grokipedia
In , the comparison of topologies on a fixed set XX refers to the partial ordering of the collection of all possible topologies on XX by set inclusion, where a topology τ2\tau_2 is said to be finer (or stronger) than another topology τ1\tau_1 if τ1⊆τ2\tau_1 \subseteq \tau_2, meaning τ2\tau_2 contains all the open sets of τ1\tau_1 and possibly more, thereby imposing a stricter notion of openness, continuity, and neighborhood structure on the . This ordering induces a coarser (or weaker) relation in the reverse direction, where τ1\tau_1 is coarser than τ2\tau_2 if every open set in τ1\tau_1 is also open in τ2\tau_2. The set of all topologies on XX forms a complete lattice under this partial order, with the discrete topology—comprising all subsets of XX—serving as the finest (maximal) element and the trivial (or indiscrete) topology—containing only the and XX itself—as the coarsest (minimal) element. Two topologies are equal if and only if they share exactly the same collection of open sets, which equivalently means they have the same closed sets or the same bases of neighborhoods at each point. Comparability is not guaranteed for every pair of topologies; if neither τ1⊆τ2\tau_1 \subseteq \tau_2 nor τ2⊆τ1\tau_2 \subseteq \tau_1, the topologies are , as illustrated by the cofinite topology on R\mathbb{R} (open sets are those with finite complements or the ) and the segment topology generated by half-open intervals (−∞,a)(-\infty, a) for a∈Ra \in \mathbb{R}, where sets like R∖{0}\mathbb{R} \setminus \{0\} belong to the former but not the latter, and vice versa. A fundamental criterion for comparison states that τ1⊆τ2\tau_1 \subseteq \tau_2 if and only if for every point x∈Xx \in X and every neighborhood UU of xx in τ1\tau_1, there exists a neighborhood VV of xx in τ2\tau_2 such that V⊆UV \subseteq U. This relation has significant implications for continuity: the identity map from (X,τ2)(X, \tau_2) to (X,τ1)(X, \tau_1) is continuous precisely when τ1\tau_1 is coarser than τ2\tau_2, allowing coarser topologies to "refine" to finer ones while preserving topological properties like or connectedness in certain directions. Such comparisons are essential in constructing specific topologies, such as and final topologies induced by families of maps, and in analyzing structures like product spaces, quotient spaces, and topological vector spaces, where relative determines convergence behaviors and separation axioms.

Fundamentals

Definition

In , the comparison of two topologies on the same underlying set XX is established through the relations of and coarseness, which are defined in terms of set inclusion. Specifically, for topologies τ1\tau_1 and τ2\tau_2 on XX, τ1\tau_1 is said to be finer than τ2\tau_2 (equivalently, τ2\tau_2 is coarser than τ1\tau_1) if τ2⊆τ1\tau_2 \subseteq \tau_1 as subsets of the power set of XX. This inclusion means that every open set in τ2\tau_2 is also open in τ1\tau_1, but the converse need not hold, allowing finer topologies to distinguish points more sharply while coarser ones impose fewer separation requirements. The relation is strict (or proper) if the inclusion is proper, i.e., τ2⊊τ1\tau_2 \subsetneq \tau_1, ensuring τ1\tau_1 has additional open sets beyond those in τ2\tau_2. A key construction in comparing topologies is the , which provides the coarsest structure compatible with a given family of maps. Given a set XX and a family of maps {fi:X→Xi}i∈I\{f_i : X \to X_i\}_{i \in I} where each XiX_i is a , the on XX is the coarsest τ\tau such that every fif_i is continuous. This is generated by the subbasis S={fi−1(Ui)∣i∈I,Ui open in Xi}\mathcal{S} = \{f_i^{-1}(U_i) \mid i \in I, U_i \text{ open in } X_i\}, where the open sets in τ\tau are arbitrary unions of finite intersections of elements from S\mathcal{S}. Dually, the on XX with respect to a family of maps {fi:Xi→X}i∈I\{f_i : X_i \to X\}_{i \in I} is the finest τ\tau making every fif_i continuous; its open sets are precisely those V⊆XV \subseteq X such that fi−1(V)f_i^{-1}(V) is open in XiX_i for all i∈Ii \in I, forming a basis consisting of such preimage-compatible sets. These notions of comparison form a partial order on the set of all topologies on XX, motivating lattice-theoretic interpretations without delving into specific operations. The concepts were formalized in the 1940s by the Bourbaki group in their foundational work on , building upon Felix Hausdorff's 1914 axiomatization of topological spaces in Grundzüge der Mengenlehre, which emphasized separation properties and laid the groundwork for abstract set-theoretic topology.

Equivalent Characterizations

In , two topologies τ1\tau_1 and τ2\tau_2 on the same set XX are comparable, with τ1\tau_1 coarser than τ2\tau_2 (or τ2\tau_2 finer than τ1\tau_1) if τ1⊆τ2\tau_1 \subseteq \tau_2 as collections of open sets. This set-theoretic inclusion admits several equivalent characterizations that facilitate verification without directly checking subset relations. These alternatives leverage dual structures like closed sets, neighborhoods, continuous maps, and subbases, providing practical criteria for comparison. The open sets criterion directly restates the inclusion: τ2\tau_2 is finer than τ1\tau_1 every in τ1\tau_1 is also open in τ2\tau_2. This follows immediately from the , as the finer contains all opens of the coarser one plus potentially more, allowing for stricter separation of points. Conversely, the closed sets dual provides an equivalent condition: every in the finer topology τ2\tau_2 is closed in the coarser topology τ1\tau_1. Since closed sets are complements of open sets, and τ1⊆τ2\tau_1 \subseteq \tau_2 implies that the closed sets of τ2\tau_2 (complements of a larger family of opens) form a of the closed sets of τ1\tau_1. This duality is fundamental, as can be axiomatized equally well via closed sets, which must include ∅\emptyset and XX, be closed under arbitrary intersections, and finite unions. A neighborhood-based characterization offers another perspective: τ2\tau_2 is finer than τ1\tau_1 , for every point x∈Xx \in X, every neighborhood of xx in the coarser topology τ1\tau_1 contains a neighborhood of xx in the finer topology τ2\tau_2. Neighborhoods of xx are sets containing an open set (in the respective ) that includes xx; in a finer , such neighborhoods tend to be smaller and more restrictive, ensuring that the coarser structure's larger neighborhoods encompass finer ones. This aligns with the axiomatic definition of topologies via neighborhood systems, where each point has a family of neighborhoods satisfying closure under supersets, finite intersections, and containing the point itself. Continuity provides a functional equivalent: assuming τ1⊆τ2\tau_1 \subseteq \tau_2, the identity map id:(X,τ2)→(X,τ1)\mathrm{id}: (X, \tau_2) \to (X, \tau_1) is continuous if and only if τ2\tau_2 is finer than τ1\tau_1. Continuity of id\mathrm{id} requires that preimages under id\mathrm{id} of open sets in τ1\tau_1 are open in τ2\tau_2, which holds precisely when every open in τ1\tau_1 is open in τ2\tau_2. More generally, for any topological space (Y,σ)(Y, \sigma), a map f:X→Yf: X \to Y that is continuous from (X,τ2)(X, \tau_2) to (Y,σ)(Y, \sigma) is also continuous from (X,τ1)(X, \tau_1) to (Y,σ)(Y, \sigma), though the converse fails unless the topologies coincide; this one-way implication underscores how finer topologies impose stronger continuity conditions. Finally, subbases yield a generative characterization: if S1\mathcal{S}_1 is a subbasis for τ1\tau_1 and S2\mathcal{S}_2 is a subbasis for τ2\tau_2 with S2⊆S1\mathcal{S}_2 \subseteq \mathcal{S}_1, then τ1\tau_1 is finer than τ2\tau_2. A subbasis S\mathcal{S} generates a whose basis consists of all finite intersections of elements from S\mathcal{S}, and the topology itself is the collection of arbitrary unions of those basis elements; inclusion of subbases ensures that the generated basis for τ2\tau_2 refines into the basis for τ1\tau_1, yielding more open sets in τ1\tau_1. This is particularly useful in constructions like product or topologies, where subbasis inclusion simplifies refinement checks.

Examples

Discrete and Indiscrete Cases

The indiscrete topology on a set XX, also known as the trivial topology, consists solely of the empty set and XX itself, forming the collection Ti={∅,X}\mathcal{T}_i = \{\emptyset, X\}. This structure satisfies the axioms of a topology vacuously for unions and finite intersections beyond these sets, making it the coarsest possible topology on XX; every other topology on XX contains Ti\mathcal{T}_i as a subset and is thus finer than it. In contrast, the discrete topology on XX is given by the power set Td=P(X)\mathcal{T}_d = \mathcal{P}(X), the collection of all subsets of XX. As the largest possible collection of subsets that forms a topology, it is the finest topology on XX, containing every other topology on XX as a subset. The discrete topology is finer than the indiscrete topology, with Ti⊂Td\mathcal{T}_i \subset \mathcal{T}_d, and this inclusion is strict unless ∣X∣≤1|X| \leq 1, in which case the two topologies coincide. A key implication of this comparison arises with singletons: in the discrete topology, every singleton {x}\{x\} for x∈Xx \in X is open, as it belongs to P(X)\mathcal{P}(X). In the indiscrete topology, however, no singleton is open unless ∣X∣=1|X| = 1, since the only nonempty open set is XX itself. To illustrate, consider the finite set X={a,b}X = \{a, b\}. The indiscrete topology on XX has open sets {∅,{a,b}}\{\emptyset, \{a, b\}\}, while the discrete topology has open sets {∅,{a},{b},{a,b}}\{\emptyset, \{a\}, \{b\}, \{a, b\}\}. The collection for the indiscrete topology is a proper subset of that for the discrete topology, confirming the strict refinement in this case.

Order Topologies

In a partially ordered set (poset) (P,≤)(P, \leq), the order topology is generated by the subbasis consisting of all open lower rays L(a)={x∈P∣x<a}L(a) = \{x \in P \mid x < a\} and open upper rays U(a)={x∈P∣a<x}U(a) = \{x \in P \mid a < x\} for each a∈Pa \in P. A basis for this topology comprises the order intervals (a,b)={x∈P∣a<x<b}(a, b) = \{x \in P \mid a < x < b\}, along with appropriate unbounded rays when the poset lacks minimal or maximal elements. The order topology is finer than the Alexandrov topology on the same poset, where the open sets are precisely the upper sets (upward-closed subsets U⊆PU \subseteq P such that if x∈Ux \in U and x≤yx \leq y, then y∈Uy \in U). Every upper set in the Alexandrov topology is a union of upper rays U(a)U(a), which are open in the order topology, so all Alexandrov-open sets are order-open; however, lower rays L(a)L(a) are generally not upper sets, yielding strictly more open sets in the order topology unless the poset is discrete. On the real numbers R\mathbb{R} equipped with the usual total order, the standard Euclidean topology coincides with the order topology, generated by the open intervals (a,b)(a, b) for a,b∈Ra, b \in \mathbb{R}. In contrast, the Sorgenfrey topology (or lower limit topology) on R\mathbb{R}, with basis consisting of half-open intervals [a,b)[a, b) for a<ba < b, is strictly finer than this order topology: every open interval (c,d)(c, d) is a union of such [a,b)[a, b) with c≤a<b≤dc \leq a < b \leq d, but sets like [0,1)[0, 1) are open in the Sorgenfrey topology yet not in the order topology. As an example of coincidence, consider the integers Z\mathbb{Z} under the usual order ≤\leq. Here, the order topology is the discrete topology, since each singleton {n}=(n−1,n+1)∩Z\{n\} = (n-1, n+1) \cap \mathbb{Z} is an open interval in the poset, making every subset open.

Properties

Continuity and Maps

In topology, the comparison of topologies on the same underlying set XX reveals a monotonic relationship with respect to continuity of functions. Suppose τ1\tau_1 is finer than τ2\tau_2 on XX (i.e., τ2⊆τ1\tau_2 \subseteq \tau_1). Then, any function f:(X,τ2)→(Y,σ)f: (X, \tau_2) \to (Y, \sigma) that is continuous with respect to τ2\tau_2 is also continuous when the domain is equipped with the finer topology τ1\tau_1, since the preimage under ff of any open set in σ\sigma belongs to the smaller collection τ2\tau_2 and hence to the larger τ1\tau_1. Conversely, equipping the domain with a finer topology makes continuity harder to achieve, as there are fewer continuous functions from (X,τ1)(X, \tau_1) to (Y,σ)(Y, \sigma) than from (X,τ2)(X, \tau_2) to (Y,σ)(Y, \sigma), because the preimages must now lie in a strictly larger family of open sets. This monotonicity extends to maps into the space as well. For functions g:(Z,ρ)→(X,τ)g: (Z, \rho) \to (X, \tau), if τ1\tau_1 is finer than τ2\tau_2 on XX, then continuity of gg with respect to τ1\tau_1 implies continuity with respect to τ2\tau_2, but the converse fails in general, resulting in more continuous maps into the coarser topology. A key construction arising from such comparisons is the initial topology, which provides a universal framework for ensuring continuity of a family of maps. Given a set XX and a family of functions {fi:X→(Yi,σi)}i∈I\{f_i: X \to (Y_i, \sigma_i)\}_{i \in I} to topological spaces, the initial topology τ\tau on XX is the coarsest topology (i.e., the one with the fewest open sets) such that each fif_i is continuous. Its universal property states that for any topological space (Z,ρ)(Z, \rho) and any function g:Z→Xg: Z \to X, gg is continuous with respect to τ\tau if and only if fi∘g:Z→Yif_i \circ g: Z \to Y_i is continuous for every i∈Ii \in I. Moreover, any topology on XX coarser than τ\tau would fail to make at least one fif_i continuous, while any finer topology would also render all fif_i continuous. Dually, the final topology addresses continuity for maps from a family into XX. For functions {fi:(Yi,σi)→X}i∈I\{f_i: (Y_i, \sigma_i) \to X\}_{i \in I}, the final topology τ\tau on XX is the finest topology (i.e., the one with the most open sets) such that each fif_i is continuous. Its universal property is that for any topological space (Z,ρ)(Z, \rho) and function g:X→Zg: X \to Z, gg is continuous with respect to τ\tau if and only if g∘fi:Yi→Zg \circ f_i: Y_i \to Z is continuous for every i∈Ii \in I. Any coarser topology on XX would make all fif_i continuous, but any finer topology would render at least one fif_i discontinuous. Representative examples illustrate these concepts in the context of topology comparisons. The subspace topology on a subset A⊆(X,τ)A \subseteq (X, \tau) is the initial topology on AA generated by the inclusion map i:A→Xi: A \to X, consisting of sets U∩AU \cap A for U∈τU \in \tau, as it is the coarsest topology making ii continuous. Similarly, for an equivalence relation ∼\sim on XX, the quotient topology on X/∼X/ \sim is the final topology generated by the projection map q:X→X/∼q: X \to X/ \sim, defined by declaring V⊆X/∼V \subseteq X/ \sim open if q−1(V)q^{-1}(V) is open in τ\tau, as it is the finest topology making qq continuous. These constructions preserve the order of topologies in the category of topological spaces. Specifically, the set of continuous maps Hom((X,τ),(Y,σ))\mathrm{Hom}((X, \tau), (Y, \sigma)) decreases (becomes a subset) as τ\tau becomes finer on XX, while it increases as σ\sigma becomes coarser on YY, reflecting the monotonicity of continuity under topology refinements.

Separation and Connectedness

In the comparison of topologies on a fixed set XX, the satisfaction of separation axioms exhibits monotonicity under refinement. If σ⊆τ\sigma \subseteq \tau (i.e., σ\sigma is coarser than the finer topology τ\tau), and (X,σ)(X, \sigma) satisfies a basic separation axiom such as T0_0 (Kolmogorov), T1_1, or T2_2 (Hausdorff), then (X,τ)(X, \tau) also satisfies it. The open sets required for separation in the coarser topology are a subset of those in the finer topology, ensuring that the distinguishing opens remain available and potentially allowing for even stronger separation. Coarsening a topology, however, risks violating these axioms, as the reduced collection of open sets may fail to distinguish points or sets that were separable before. The extreme cases highlight this behavior clearly. The discrete topology on XX, being the finest possible (with all subsets open), satisfies every separation axiom Tn_n for arbitrary n≥0n \geq 0, as singletons are open and can separate any finite collection of points or closed sets. Conversely, the indiscrete topology (with only ∅\emptyset and XX open), the coarsest possible, satisfies even the weakest T0_0 axiom only if ∣X∣≤1|X| \leq 1; for ∣X∣>1|X| > 1, no proper nonempty open sets exist to separate distinct points. Connectedness follows the opposite monotonicity. If σ⊆τ\sigma \subseteq \tau and (X,τ)(X, \tau) is connected, then (X,σ)(X, \sigma) is connected, since any disconnection of (X,σ)(X, \sigma) into two nonempty disjoint open sets whose union is XX would use opens from σ\sigma (hence also from τ\tau), contradicting the connectedness of (X,τ)(X, \tau). Coarser topologies thus admit fewer potential disconnections and are inherently more prone to connectedness. Refinement can destroy connectedness by introducing additional open sets that enable nontrivial separations of XX. Path-connectedness, a stricter condition requiring continuous paths between any pair of points, behaves analogously. If σ⊆τ\sigma \subseteq \tau and (X,τ)(X, \tau) is path-connected, then (X,σ)(X, \sigma) is path-connected: any path [0,1]→X[0,1] \to X continuous with respect to τ\tau remains continuous with respect to σ\sigma, as preimages of the fewer opens in σ\sigma are among those already open in the domain under the continuity for τ\tau. Coarsening preserves path-connectedness, but refinement may eliminate paths by making some maps discontinuous. An illustrative example is the rational numbers Q\mathbb{Q} under the subspace topology induced from the usual topology on R\mathbb{R}, which is coarser than the discrete topology on Q\mathbb{Q}. This subspace topology renders Q\mathbb{Q} totally disconnected—its only connected subsets are singletons—yet it is not discrete, as singletons are not open. The discrete topology on Q\mathbb{Q} is likewise totally disconnected, demonstrating that coarsening from the discrete case preserves total disconnectedness here.

Lattice Structure

Poset Framework

The set of all topologies on a fixed set XX, denoted T(X)\mathcal{T}(X), forms a partially ordered set (poset) under the inclusion relation ⊆\subseteq, where τ≤σ\tau \leq \sigma if and only if τ⊆σ\tau \subseteq \sigma. In this ordering, τ\tau is coarser than σ\sigma (or equivalently, σ\sigma is finer than τ\tau), meaning that every open set in τ\tau is also open in σ\sigma, but not necessarily vice versa. This poset structure captures the hierarchical relationships among topologies, allowing for systematic comparisons of their refinement levels. The poset (T(X),⊆)(\mathcal{T}(X), \subseteq) is in fact a complete lattice, bounded below by the indiscrete topology (containing only ∅\emptyset and XX) and bounded above by the discrete topology (containing all subsets of XX). For any subset {τi∣i∈I}⊆T(X)\{\tau_i \mid i \in I\} \subseteq \mathcal{T}(X), the infimum is the intersection ⋂i∈Iτi\bigcap_{i \in I} \tau_i, which is itself a topology, and the supremum is the topology generated by the union ⋃i∈Iτi\bigcup_{i \in I} \tau_i, consisting of all arbitrary unions of finite intersections of sets from the union. This lattice structure ensures that every collection of topologies has both a greatest lower bound and a least upper bound within T(X)\mathcal{T}(X). The dual poset (T(X),⊇)(\mathcal{T}(X), \supseteq) reverses the order, such that τ≤σ\tau \leq \sigma now means τ⊇σ\tau \supseteq \sigma, thereby swapping the notions of coarser and finer: what was finer becomes "smaller" in this antitone ordering. This duality highlights the symmetry in topological comparisons and is useful in contexts like duality theory for topological spaces. The lattice is chain-complete: for any chain {τα∣α∈A}\{\tau_\alpha \mid \alpha \in A\} (a totally ordered subset under ⊆\subseteq), the supremum exists and is the generated by ⋃α∈Aτα\bigcup_{\alpha \in A} \tau_\alpha. Note that the bare union itself may not form a , as it might fail closure under arbitrary unions, but the generated does serve as the least upper bound. The poset of all Hausdorff topologies on XX without isolated points (ordered by ⊆\subseteq) is inductive—every chain has an upper bound given by the supremum as above—and thus admits maximal elements by Zorn's lemma, corresponding to maximal such topologies that cannot be properly refined without introducing isolated points. Similar applications of Zorn's lemma exist for other properties not preserved under refinement.

Meet and Join Operations

In the lattice of topologies on a fixed set XX, the meet operation, or infimum, of an arbitrary family of topologies {τi}i∈I\{\tau_i\}_{i \in I} is defined as their intersection ⋂i∈Iτi\bigcap_{i \in I} \tau_i. This intersection is always a topology, as it satisfies the axioms of being closed under arbitrary unions and finite intersections, and it represents the finest topology coarser than each member of the family (the greatest lower bound), containing precisely those sets open in every τi\tau_i. The join operation, or supremum, of the same family is the coarsest topology containing ⋃i∈Iτi\bigcup_{i \in I} \tau_i (making all those sets open), constructed by taking all arbitrary unions of finite intersections of elements from the union; this yields the coarsest topology finer than each τi\tau_i (the least upper bound). For the binary case, the meet of two topologies τ1\tau_1 and τ2\tau_2 simplifies to their set-theoretic intersection τ1∧τ2=τ1∩τ2\tau_1 \wedge \tau_2 = \tau_1 \cap \tau_2, which remains a topology coarser than both. The join τ1∨τ2\tau_1 \vee \tau_2 is the topology generated by τ1∪τ2\tau_1 \cup \tau_2 as a subbasis, finer than each and the least upper bound in the lattice order. These operations exhibit , as τ∨τ=τ\tau \vee \tau = \tau for any τ\tau, since the topology generated by τ∪τ=τ\tau \cup \tau = \tau is τ\tau itself; similarly, τ∧τ=τ\tau \wedge \tau = \tau. The indiscrete topology {∅,X}\{\emptyset, X\}, as the bottom element of the lattice, absorbs any topology from below via the meet: {∅,X}∧τ={∅,X}\{\emptyset, X\} \wedge \tau = \{\emptyset, X\}. Conversely, the discrete topology 2X2^X, the top element, absorbs from above via the join: τ∨2X=2X\tau \vee 2^X = 2^X. However, the lattice of topologies fails to be distributive in general when ∣X∣≥3|X| \geq 3. A counterexample on the three-point set X={a,b,c}X = \{a, b, c\} involves the topologies τ1={∅,X,{b},{a,c}}\tau_1 = \{\emptyset, X, \{b\}, \{a, c\}\}, τ2={∅,X,{a},{b,c}}\tau_2 = \{\emptyset, X, \{a\}, \{b, c\}\}, and τ3={∅,X,{b},{a,b}}\tau_3 = \{\emptyset, X, \{b\}, \{a, b\}\}. Here, τ1∨(τ2∧τ3)={∅,X,{b},{a,c}}\tau_1 \vee (\tau_2 \wedge \tau_3) = \{\emptyset, X, \{b\}, \{a, c\}\}, while (τ1∨τ2)∧(τ1∨τ3)={∅,X,{a},{b},{a,b},{a,c}}(\tau_1 \vee \tau_2) \wedge (\tau_1 \vee \tau_3) = \{\emptyset, X, \{a\}, \{b\}, \{a, b\}, \{a, c\}\}, violating the distributive law τ1∨(τ2∧τ3)=(τ1∨τ2)∧(τ1∨τ3)\tau_1 \vee (\tau_2 \wedge \tau_3) = (\tau_1 \vee \tau_2) \wedge (\tau_1 \vee \tau_3).

References

  1. http://www.math.uni-konstanz.de/~infusino/TVS-WS18-19/Lect2.pdf
  2. https://groupoids.org.uk/pdffiles/topgrpds-e.pdf
  3. https://archive.org/details/generaltopology0000bour
  4. https://planetmath.org/initialtopology
  5. https://www.rexresearch1.com/TopologyLibrary/GeneralTopologyWillard.pdf
  6. https://math.berkeley.edu/~wodzicki/H104.F10/Topology.pdf
  7. https://www.math.toronto.edu/ivan/mat327/docs/notes/01-intro.pdf
  8. https://math.jhu.edu/~eriehl/161/topologies.pdf
  9. https://www.math.uh.edu/~almus/Topology_after.pdf
  10. https://math.berkeley.edu/~shiyu/s15capstone/materials/Capstone_Topology.pdf
  11. https://www.math.toronto.edu/ivan/mat327/docs/lecturenotes/lecture2-AsadDurrani.pdf
  12. https://www.math.colostate.edu/~clayton/courses/500/500_midterm.pdf
  13. https://math.stackexchange.com/questions/3361867/what-is-definition-of-order-topology
  14. https://faculty.etsu.edu/gardnerr/5357/notes/Munkres-14.pdf
  15. https://math.stackexchange.com/questions/422893/partial-order-induced-topology
  16. https://faculty.etsu.edu/gardnerr/5357/notes/Munkres-13.pdf
  17. https://ece.iisc.ac.in/~parimal/2015/proofs/lecture-14.pdf
  18. https://mathweb.ucsd.edu/~jmckerna/Teaching/24-25/Winter/190A/l_8.pdf
  19. https://www.math.uh.edu/~haynes/files/topgps3.pdf
  20. https://davidaltizio.web.illinois.edu/alg_topo_notes.pdf
  21. https://web.math.arizona.edu/files/grad/workshops/integration/_Topology_Problems_2016.pdf
  22. https://mathweb.ucsd.edu/~bprhoades/190w18/quotient.pdf
  23. https://arxiv.org/pdf/1908.09366
  24. https://scholarworks.gvsu.edu/cgi/viewcontent.cgi?article=1030&context=books
  25. https://web.ma.utexas.edu/ibl1/courses/resources/12_4_2004_undergrad_top_mooremethod.pdf
  26. http://math.bu.edu/people/mabeck/Autumn11/tutorial_sheet_6_wsoln.pdf
  27. https://ncatlab.org/nlab/show/finer%2Btopology
  28. https://arxiv.org/abs/2508.05419
  29. https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-5/issue-2/The-lattice-of-topologies-A-survey-equations/10.1216/RMJ-1975-5-2-177.pdf
  30. https://mathoverflow.net/questions/484088/is-there-a-natural-topology-for-sets-of-topological-spaces
  31. https://mathoverflow.net/questions/313626/is-the-topology-generated-by-the-union-of-a-chain-of-paracompact-topologies-para
  32. https://msp.org/pjm/1979/84-1/pjm-v84-n1-p05-s.pdf
  33. https://mathoverflow.net/questions/330710/whats-the-minimal-weight-of-a-maximal-space
  34. https://mathoverflow.net/questions/109007/constructible-sets-in-hausdorff-spaces
Add your contribution
Related Hubs
User Avatar
No comments yet.