Base (topology)

In mathematics, a base or basis for the topology $τ$ of a topological space $(X, τ)$ is a family $B$ of open subsets of $X$ such that every open set is equal to a union of some sub-family of $B$ [1][2][3][4][5] (this sub-family is allowed to be infinite, finite, or even empty[note 1]). For example, the set of all open intervals in the real number line $\mathbb {R}$ is a basis for the Euclidean topology on $\mathbb {R}$ because every open interval is an open set, and also every open subset of $\mathbb {R}$ can be written as a union of some family of open intervals.

Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called basic open sets, are often easier to describe and use than arbitrary open sets.[6] Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.

Not all families of subsets form a base for a topology. For example, because $X$ is always an open subset of every topology on $X$ , if a family $B$ of subsets is to be a base for a topology on $X$ then it must cover $X$ , which by definition means that the union of all sets in $B$ must be equal to $X$ . If $X$ has more than one point then there exist families of subsets of $X$ that do not cover $X$ and consequently, they can not form a basis for any topology on $X$ . A family $B$ of subsets of $X$ that does form a basis for some topology on $X$ is called a base for a topology on $X$ ,[1][2][3] in which case this necessarily unique topology, call it $τ$ , is said to be generated by $B$ and $B$ is consequently a basis for the topology $τ$ . Such families of sets are frequently used to define topologies. A weaker notion related to bases is that of a subbasis for a topology. Bases for topologies are closely related to neighborhood bases.

Definition and basic properties

A base is a collection B of open subsets of X satisfying the following properties:

The base elements cover X.
Let B₁, B₂ be base elements and let I be their intersection. Then for each x in I, there is a base element B₃ containing x such that B₃ is subset of I.

An equivalent property is: any finite intersection[note 2] of elements of B can be written as a union of elements of B. These two conditions are exactly what is needed to ensure that the set of all unions of subsets of B is a topology on X.

If a collection B of subsets of X fails to satisfy these properties, then it is not a base for any topology on X. (It is a subbase, however, as is any collection of subsets of X.) Conversely, if B satisfies these properties, then there is a unique topology on X for which B is a base; it is called the topology generated by B. (This topology is the intersection of all topologies on X containing B.) This is a very common way of defining topologies. A sufficient but not necessary condition for B to generate a topology on X is that B is closed under intersections; then we can always take B₃ = I above.

For example, the collection of all open intervals in the real line forms a base for a topology on the real line because the intersection of any two open intervals is itself an open interval or empty. In fact they are a base for the standard topology on the real numbers.

However, a base is not unique. Many different bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals. In contrast to a basis of a vector space in linear algebra, a base need not be maximal; indeed, the only maximal base is the topology itself. In fact, any open set generated by a base may be safely added to the base without changing the topology. The smallest possible cardinality of a base is called the weight of the topological space.

An example of a collection of open sets which is not a base is the set S of all semi-infinite intervals of the forms (−∞, a) and (a, ∞), where a is a real number. Then S is not a base for any topology on R. To show this, suppose it were. Then, for example, (−∞, 1) and (0, ∞) would be in the topology generated by S, being unions of a single base element, and so their intersection (0,1) would be as well. But (0, 1) clearly cannot be written as a union of elements of S. Using the alternate definition, the second property fails, since no base element can "fit" inside this intersection.

Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.

Examples

The set $Γ$ of all open intervals in $ℝ$ form a basis for the Euclidean topology on $ℝ$ . Every topology $τ$ on a set $X$ is a basis for itself (that is, $τ$ is a basis for $τ$ ). Because of this, if a theorem's hypotheses assumes that a topology $τ$ has some basis $Γ$ , then this theorem can be applied using $Γ := τ$ .

A non-empty family of subsets of a set $X$ that is closed under finite intersections of two or more sets, which is called a $π$ -system on $X$ , is necessarily a base for a topology on $X$ if and only if it covers $X$ . By definition, every σ-algebra, every filter (and so in particular, every neighborhood filter), and every topology is a covering $π$ -system and so also a base for a topology. In fact, if $Γ$ is a filter on $X$ then ${ \emptyset } \cup Γ$ is a topology on $X$ and $Γ$ is a basis for it. A base for a topology does not have to be closed under finite intersections and many aren't. But nevertheless, many topologies are defined by bases that are also closed under finite intersections. For example, each of the following families of subset of $ℝ$ is closed under finite intersections and so each forms a basis for some topology on $ℝ$ :

The set $Γ$ of all bounded open intervals in $ℝ$ generates the usual Euclidean topology on $ℝ$ .
The set $Σ$ of all bounded closed intervals in $ℝ$ generates the discrete topology on $ℝ$ and so the Euclidean topology is a subset of this topology. This is despite the fact that $Γ$ is not a subset $Σ$ . Consequently, the topology generated by $Γ$ , which is the Euclidean topology on $ℝ$ , is coarser than the topology generated by $Σ$ . In fact, it is strictly coarser because $Σ$ contains non-empty compact sets which are never open in the Euclidean topology.
The set $Γ ℚ$ of all intervals in $Γ$ such that both endpoints of the interval are rational numbers generates the same topology as $Γ$ . This remains true if each instance of the symbol $Γ$ is replaced by $Σ$ .
$Σ \infty = { [r, \infty) : r \in ℝ }$ generates a topology that is strictly coarser than the topology generated by $Σ$ . No element of $Σ \infty$ is open in the Euclidean topology on $ℝ$ .
$Γ \infty = { (r, \infty) : r \in ℝ }$ generates a topology that is strictly coarser than both the Euclidean topology and the topology generated by $Σ \infty$ . The sets $Σ \infty$ and $Γ \infty$ are disjoint, but nevertheless $Γ \infty$ is a subset of the topology generated by $Σ \infty$ .

Objects defined in terms of bases

The order topology is usually defined as the topology generated by a collection of open-interval-like sets.
The metric topology is usually defined as the topology generated by a collection of open balls.
A second-countable space is one that has a countable base.
The discrete topology has the singletons as a base.
The profinite topology on a group is defined by taking the collection of all normal subgroups of finite index as a basis of open neighborhoods of the identity.

The Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties. For the usual basis of this topology, every finite intersection of basis elements is a basis element. Therefore bases are sometimes required to be stable by finite intersection.

The Zariski topology of $\mathbb {C} ^{n}$ is the topology that has the algebraic sets as closed sets. It has a basis formed by the set complements of algebraic hypersurfaces.
The Zariski topology of the spectrum of a ring (the set of the prime ideals) has a basis such that each element consists of all prime ideals that do not contain a given element of the ring.

Theorems

For each point x in an open set U, there is a base element containing x and contained in U.
A topology T₂ is finer than a topology T₁ if and only if for each x and each base element B of T₁ containing x, there is a base element of T₂ containing x and contained in B.
If B₁,B₂,...,B_n are bases for the topologies T₁,T₂,...,T_n, then the set product B₁ × B₂ × ... × B_n is a base for the product topology T₁ × T₂ × ... × T_n. In the case of an infinite product, this still applies, except that all but finitely many of the base elements must be the entire space.
Let B be a base for X and let Y be a subspace of X. Then if we intersect each element of B with Y, the resulting collection of sets is a base for the subspace Y.
If a function f : X → Y maps every base element of X into an open set of Y, it is an open map. Similarly, if every preimage of a base element of Y is open in X, then f is continuous.
A collection of subsets of X is a topology on X if and only if it generates itself.
B is a basis for a topological space X if and only if the subcollection of elements of B which contain x form a local base at x, for any point x of X.

Base for the closed sets

Closed sets are equally adept at describing the topology of a space. There is, therefore, a dual notion of a base for the closed sets of a topological space. Given a topological space X, a family of closed sets F forms a base for the closed sets if and only if for each closed set A and each point x not in A there exists an element of F containing A but not containing x.

It is easy to check that F is a base for the closed sets of X if and only if the family of complements of members of F is a base for the open sets of X.

Let F be a base for the closed sets of X. Then

∩F = ∅
For each F₁ and F₂ in F the union F₁ ∪ F₂ is the intersection of some subfamily of F (i.e. for any x not in F₁ or F₂ there is an F₃ in F containing F₁ ∪ F₂ and not containing x).

Any collection of subsets of a set X satisfying these properties forms a base for the closed sets of a topology on X. The closed sets of this topology are precisely the intersections of members of F.

In some cases it is more convenient to use a base for the closed sets rather than the open ones. For example, a space is completely regular if and only if the zero sets form a base for the closed sets. Given any topological space X, the zero sets form the base for the closed sets of some topology on X. This topology will be the finest completely regular topology on X coarser than the original one. In a similar vein, the Zariski topology on Aⁿ is defined by taking the zero sets of polynomial functions as a base for the closed sets.

Weight and character

We shall work with notions established in (Engelking 1977, p. 12, pp. 127-128).

Fix X a topological space. Here, a network is a family ${\mathcal {N}}$ of sets, for which, for all points x and open neighbourhoods U containing x, there exists B in ${\mathcal {N}}$ for which x ∈ B ⊆ U. Note that, unlike a basis, the sets in a network need not be open.

We define the weight, w(X), as the minimum cardinality of a basis; we define the network weight, nw(X), as the minimum cardinality of a network; the character of a point, $\chi (x,X)$ , as the minimum cardinality of a neighbourhood basis for x in X; and the character of X to be

\chi (X)\triangleq \sup\{\chi (x,X):x\in X\}.

The point of computing the character and weight is to be able to tell what sort of bases and local bases can exist. We have the following facts:

nw(X) ≤ w(X).
if X is discrete, then w(X) = nw(X) = |X|.
if X is Hausdorff, then nw(X) is finite iff X is finite discrete.
if B is a basis of X then there is a basis $B'\subseteq B$ of size $|B'|\leq w(X)$ .
if N a neighbourhood basis for x in X then there is a neighbourhood basis $N'\subseteq N$ of size $|N'|\leq \chi (x,X)$ .
if f : X → Y is a continuous surjection, then nw(Y) ≤ w(X). (Simply consider the Y-network $f'''B\triangleq \{f''U:U\in B\}$ for each basis B of X.)
if $(X,\tau )$ is Hausdorff, then there exists a weaker Hausdorff topology $(X,\tau ')$ so that $w(X,\tau ')\leq nw(X,\tau )$ . So a fortiori, if X is also compact, then such topologies coincide and hence we have, combined with the first fact, nw(X) = w(X).
if f : X → Y a continuous surjective map from a compact metrisable space to an Hausdorff space, then Y is compact metrisable.

The last fact follows from f(X) being compact Hausdorff, and hence $nw(f(X))=w(f(X))\leq w(X)\leq \aleph _{0}$ (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable. (An application of this, for instance, is that every path in an Hausdorff space is compact metrisable.)

Increasing chains of open sets

Using the above notation, suppose that w(X) ≤ κ some infinite cardinal. Then there does not exist a strictly increasing sequence of open sets (equivalently strictly decreasing sequence of closed sets) of length ≥ κ⁺.

To see this (without the axiom of choice), fix

\left\{U_{\xi }\right\}_{\xi \in \kappa },

as a basis of open sets. And suppose per contra, that

\left\{V_{\xi }\right\}_{\xi \in \kappa ^{+}}

were a strictly increasing sequence of open sets. This means

\forall \alpha <\kappa ^{+}:\qquad V_{\alpha }\setminus \bigcup _{\xi <\alpha }V_{\xi }\neq \varnothing .

For

x\in V_{\alpha }\setminus \bigcup _{\xi <\alpha }V_{\xi },

we may use the basis to find some U_γ with x in U_γ ⊆ V_α. In this way we may well-define a map, f : κ⁺ → κ mapping each α to the least γ for which U_γ ⊆ V_α and meets

V_{\alpha }\setminus \bigcup _{\xi <\alpha }V_{\xi }.

This map is injective, otherwise there would be α < β with f(α) = f(β) = γ, which would further imply U_γ ⊆ V_α but also meets

V_{\beta }\setminus \bigcup _{\xi <\alpha }V_{\xi }\subseteq V_{\beta }\setminus V_{\alpha },

which is a contradiction. But this would go to show that κ⁺ ≤ κ, a contradiction.

Notes

By a standard convention, the empty set, which is always open, is the union of the empty collection.
We are using a convention that the empty intersection of subsets of X is considered finite and is equal to X.

References

Bourbaki 1989, pp. 18-21.
Dugundji 1966, pp. 62-68.
Willard 2004, pp. 37-40.
Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons. p. 16. ISBN 0-471-83817-9. Retrieved 27 July 2012. Definition. A collection B of open subsets of a topological space (X,T) is called a basis for T if every open set can be expressed as a union of members of B.
Armstrong, M. A. (1983). Basic Topology. Springer. p. 30. ISBN 0-387-90839-0. Retrieved 13 June 2013. Suppose we have a topology on a set $X$ , and a collection $\beta$ of open sets such that every open set is a union of members of $\beta$ . Such a family of open sets is said to generate or define this topology. Then $\beta$ is called a base for the topology...
Adams & Franzosa 2009, pp. 46-56.

Bibliography

Adams, Colin; Franzosa, Robert (2009). Introduction to Topology: Pure and Applied. New Delhi: Pearson Education. ISBN 978-81-317-2692-1. OCLC 789880519.
Arkhangel’skij, A.V.; Ponomarev, V.I. (1984). Fundamentals of general topology: problems and exercises. Mathematics and Its Applications. 13. Translated from the Russian by V. K. Jain. Dordrecht: D. Reidel Publishing. Zbl 0568.54001.
Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063.
Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.
Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
Engelking, Ryszard (1977). General Topology. Monografie Matematyczne. 60. Warsaw: PWN. Zbl 0373.54002.
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Willard, Stephen (1970) General Topology. Addison-Wesley. Reprinted 2004, Dover Publications.

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[6] By a standard convention, the empty set, which is always open, is the union of the empty collection.

[8] We are using a convention that the empty intersection of subsets of X is considered finite and is equal to X.

[FOOTNOTEBourbaki198918-21-1] Bourbaki 1989, pp. 18-21.

[FOOTNOTEDugundji196662-68-2] Dugundji 1966, pp. 62-68.

[FOOTNOTEWillard200437-40-3] Willard 2004, pp. 37-40.

[4] Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons. p. 16. ISBN 0-471-83817-9. Retrieved 27 July 2012. Definition. A collection B of open subsets of a topological space (X,T) is called a basis for T if every open set can be expressed as a union of members of B.

[5] Armstrong, M. A. (1983). Basic Topology. Springer. p. 30. ISBN 0-387-90839-0. Retrieved 13 June 2013. Suppose we have a topology on a set $X$ , and a collection $\beta$ of open sets such that every open set is a union of members of $\beta$ . Such a family of open sets is said to generate or define this topology. Then $\beta$ is called a base for the topology...

[FOOTNOTEAdamsFranzosa200946-56-7] Adams & Franzosa 2009, pp. 46-56.