Subbase

In topology, a subbase (also called a subbasis, prebase, or prebasis) for the topology \tau on a topological space (X, \tau) is a subcollection \mathcal{S} \subseteq \tau such that the finite intersections of elements of \mathcal{S} form a basis for \tau. Equivalently, \tau is the smallest topology on X containing \mathcal{S}, i.e., the intersection of all topologies on X that include \mathcal{S}.^[1]^[2]

Core Concepts

Definition

A basis for a topology on a set X is a collection \mathcal{B} of subsets of X such that \bigcup_{B \in \mathcal{B}} B = X, and whenever B_1, B_2 \in \mathcal{B} and x \in B_1 \cap B_2, there exists B_3 \in \mathcal{B} with x \in B_3 \subseteq B_1 \cap B_2. The topology generated by \mathcal{B} consists of the empty set, X, and all arbitrary unions of elements from \mathcal{B}.^[3] A subbasis \mathcal{S} (or subbase) for a topology on X is a collection of subsets of X such that \bigcup_{S \in \mathcal{S}} S = X. Unlike a basis, \mathcal{S} need not satisfy the intersection condition directly; instead, the collection of all finite intersections of elements from \mathcal{S} (including the empty intersection, taken as X) forms a basis for the topology. This ensures that every point in X is covered and that the intersection property holds for the derived basis.^[4]^[5] The topology \tau(\mathcal{S}) generated by the subbasis \mathcal{S} is defined as the collection of all arbitrary unions of finite intersections of members of \mathcal{S}, along with the empty set. Formally, an open set U \in \tau(\mathcal{S}) if U = \emptyset or U = \bigcup_{i \in I} \left( \bigcap_{j=1}^{n_i} S_{i j} \right), where I is an index set, each n_i < \infty, and each S_{i j} \in \mathcal{S}. This construction yields the smallest topology on X containing \mathcal{S} as open sets, making subbases a fundamental tool for specifying topologies via simpler generating collections.^[4]^[5]

Relation to Bases and Topologies

In topology, a subbase \mathcal{S} for a topological space (X, \tau) relates to a base \mathcal{B} by generating \mathcal{B} as the collection of all finite intersections of elements from \mathcal{S}, where \mathcal{B} covers X and satisfies the base condition: for any open set U \in \tau and point x \in U, there exists B \in \mathcal{B} such that x \in B \subseteq U.^[6]^[2] This distinguishes subbases from bases, as subbases need not themselves form a base but instead produce one through this intersection process, allowing for a coarser initial collection.^[7] Subbases simplify the definition of a topology by permitting the use of a potentially smaller or less refined family of sets compared to a full base, which is particularly useful when the finite intersections of the subbase elements are sufficient to generate the required basis without needing every open set's building blocks upfront.^[6]^[2] The topology generated by a subbase \mathcal{S}, denoted \tau(\mathcal{S}), consists of all arbitrary unions of elements from the base \mathcal{B} formed by finite intersections of \mathcal{S}.^[7] For \mathcal{S} to qualify as a subbase on a set X, it must be a subset of the power set \mathcal{P}(X) with \bigcup \mathcal{S} = X, ensuring that the generated topology includes X as an open set.^[2]

Properties and Construction

Key Properties

A subbase \mathcal{S} for a topology on a set X must satisfy the coverage property, whereby the union of all elements in \mathcal{S} equals X. This condition ensures that every point in X is contained in at least one subbase element, allowing the generated topology to encompass the entire space without isolated points outside the open sets.^[8] The formation of open sets from a subbase relies on finite intersections: the associated base consists of all finite intersections of members of \mathcal{S}, and open sets are arbitrary unions thereof. Infinite intersections of subbase elements, however, are not necessarily open in the generated topology, distinguishing subbases from filters or other structures that may close under infinite operations.^[9] If \mathcal{S} \subset \mathcal{S}' where \mathcal{S}' is also a subbase for the space, the topology generated by \mathcal{S}' is finer than or equal to the topology generated by \mathcal{S}. This refinement property arises because the larger collection \mathcal{S}' imposes more open sets in the minimal topology containing it, resulting in a topology at least as fine as the one from the smaller collection.^[6] For a given topology \tau on X, the collection of all non-empty open sets in \tau serves as a subbase generating \tau, as finite intersections of open sets remain open and arbitrary unions reproduce all open sets; this collection is maximal among subbases for \tau.^[9]

Generating a Topology from a Subbase

Given a subbase S for a topology on a set X, where S is a collection of subsets of X such that \bigcup_{U \in S} U = X, the topology \tau(S) generated by S is constructed through a two-step process that ensures S generates the coarsest topology containing all elements of S.^[10]^[11] The first step involves forming a base B from S by taking all non-empty finite intersections of elements from S, including the empty intersection, which is defined as X. Formally,

B = \{ \bigcap_{i=1}^n U_i \mid n \in \mathbb{N}, \, U_i \in S \ \forall i = 1, \dots, n, \, \bigcap_{i=1}^n U_i \neq \emptyset \} \cup \{ X \},

where the convention \bigcap_{i=1}^0 U_i = X accounts for the empty intersection. This collection B serves as a base because the union of its elements covers X (since S covers X and X \in B), and it satisfies the base condition: for any B_1, B_2 \in B and x \in B_1 \cap B_2, there exists B_3 \in B such that x \in B_3 \subseteq B_1 \cap B_2, as B_1 \cap B_2 is itself a finite intersection of elements from S and thus in B.^[10]^[11]^[12] The second step constructs the topology \tau(S) as the collection of all arbitrary unions of elements from B, together with the empty set:

\tau(S) = \{ \bigcup_{j \in J} V_j \mid J \subseteq I, \, V_j \in B \} \cup \{ \emptyset \},

where I indexes the unions. This ensures \tau(S) is closed under arbitrary unions and finite intersections, making it a valid topology, and it is the smallest such topology containing S since every open set is a union of basis elements derived from S. The finite intersection property of subbases, where intersections of finitely many elements from S remain in the generated topology, supports this construction without requiring additional refinements.^[10]^[11]^[12] To verify that B indeed forms a base for \tau(S), one confirms coverage by noting X \in B and the subadditivity of unions, and the intersection condition holds because any intersection of basis elements is itself a basis element due to the finite intersection closure of S. For instance, consider the subbase S = \{ \{x\} \mid x \in X \} on a set X; the finite intersections yield B consisting of X and all singletons \{x\} (intersections of one element), generating the discrete topology \tau(S) where every subset of X is open, as arbitrary unions of singletons produce all subsets. This minimal example illustrates how the process yields the finest possible topology from point-separating subbase elements.^[10]^[11]

Examples

Basic Examples

One basic example of a subbase is the trivial subbase S = \{ X \} on a set X, where the finite intersections of elements from S consist solely of X itself (considering the empty intersection as X by convention). The topology generated by this subbase is the indiscrete topology \{ \emptyset, X \}, as all unions of these intersections yield only the empty set and X.^[13] Another foundational example is the discrete subbase S = \{ \{x\} \mid x \in X \}, the collection of all singleton subsets of X. Finite intersections of these singletons are either singletons (if all are the same) or empty (if distinct), and the generated topology is the discrete topology on X, where every subset is open, since arbitrary unions of singletons produce all possible subsets. This collection serves as both a base and a subbase for the discrete topology.^[14] A standard subbase for the Euclidean topology on the real line \mathbb{R} is the collection S = \{ (-\infty, a) \mid a \in \mathbb{R} \} \cup \{ (b, \infty) \mid b \in \mathbb{R} \} of all open rays. Finite intersections of these rays are open intervals of the form (c, d) (or rays or \mathbb{R}), which form a base, and arbitrary unions generate all open sets in the standard topology. This illustrates the use of unbounded sets to generate bounded basis elements through intersections.^[8] For the cofinite topology on an infinite set X, the collection S of all cofinite subsets (those with finite complement) forms a subbase. Finite intersections of cofinite sets remain cofinite (or X), providing a base of cofinite sets, and the generated topology has as open sets precisely the empty set, X, and all cofinite subsets. This example highlights a subbase that coincides with its derived base in coarser topologies.^[15]

Examples in Common Topological Spaces

In Euclidean space \mathbb{R}^n equipped with the standard topology, a subbase consists of all open half-spaces defined by coordinate inequalities, specifically the sets \{ x \in \mathbb{R}^n \mid x_i > a \} and \{ x \in \mathbb{R}^n \mid x_i < b \} for each coordinate index i = 1, \dots, n and all real numbers a, b.^[13] Finite intersections of these half-spaces yield open rectangles, which are products of open intervals in each coordinate and form a base for the standard Euclidean topology.^[13] This construction arises naturally from viewing \mathbb{R}^n as the product of n copies of \mathbb{R}, where the subbase elements correspond to cylindrical regions unbounded in all but one direction.^[8] In a general metric space (X, d), the collection of all open balls of a fixed radius r > 0, namely \{ B(x, r) \mid x \in X \}, serves as a subbase for the metric topology.^[8] Finite intersections of these fixed-radius balls produce sets that effectively refine to smaller neighborhoods through the choice of centers, enabling the generated base to approximate open balls of arbitrary smaller radii via the triangle inequality.^[16] The resulting topology coincides with the standard metric topology, as every open set can be expressed as a union of such finite intersections.^[8] For the order topology on a totally ordered set X, the subbase is formed by the collection of all open rays (a, +\infty) and (-\infty, b) where a, b \in X.^[17] Finite intersections of these rays yield open intervals of the form (a, b), which constitute a base for the order topology.^[17] This subbase captures the natural ordering structure, ensuring that the generated topology respects the linear order on X.^[17] The Sierpiński space, with underlying set \{0, 1\} and topology \{\emptyset, \{0\}, \{0,1\}\}, can be generated by the subbase \mathcal{S} = \{\{0\}, \{0,1\}\}.^[18] The finite intersections of elements from \mathcal{S} are \{0,1\} (empty intersection or single \{0,1\}), \{0\} (intersection of \{0\} with itself or with \{0,1\}), and \emptyset (if considering improper cases, though typically excluded), forming a base whose arbitrary unions produce the full topology.^[18] This minimal example illustrates a non-Hausdorff space where the subbase directly encodes the asymmetric openness of the points.^[18]

Theorems and Applications

Alexander Subbase Theorem

The Alexander subbase theorem provides a criterion for compactness using subbases. Let X be a topological space with subbase \mathcal{B}. Then X is compact if and only if every open cover of X by elements of \mathcal{B} has a finite subcover.^[19] This theorem, due to James Waddell Alexander II, is equivalent to the standard definition of compactness and is particularly useful in proving Tychonoff's theorem for products of compact spaces, as the product topology has a natural subbase from the factor topologies. The proof relies on the axiom of choice and shows that the condition implies every open cover (not just subbasic ones) has a finite subcover, since any base can be refined from subbasic covers. Conversely, compactness implies the subbasic cover condition.^[19]

Subbases in Product and Initial Topologies

In the product topology on the Cartesian product \prod_{i \in I} X_i of topological spaces (X_i, \tau_i), a subbase is given by the collection \mathcal{S} = \bigcup_{i \in I} \{\pi_i^{-1}(U_i) \mid U_i \in \tau_i\}, where \pi_i: \prod_{j \in I} X_j \to X_i is the projection map onto the i-th coordinate.^[20] The finite intersections of elements from this subbase form a base consisting of sets of the form \prod_{i \in I} U_i, where each U_i \in \tau_i and U_i = X_i for all but finitely many i.^[20] This construction ensures that the product topology is the coarsest topology making all projection maps \pi_i continuous.^[20] The initial topology on a set X with respect to a family of maps \{f_\alpha: X \to Y_\alpha \mid \alpha \in A\}, where each (Y_\alpha, \sigma_\alpha) is a topological space, is defined by the subbase \mathcal{S} = \{f_\alpha^{-1}(V_\alpha) \mid V_\alpha \in \sigma_\alpha, \alpha \in A\}.^[21] This generates the coarsest topology on X such that every map f_\alpha is continuous, as any topology rendering the f_\alpha continuous must contain \mathcal{S} as open sets, and the topology generated by \mathcal{S} is the smallest such collection.^[21] The product topology is a special case of the initial topology, where the maps are the projections \pi_i.^[21] In function spaces, the initial topology construction applies naturally to the space \mathbb{R}^X of all functions from a set X to \mathbb{R}, equipped with the pointwise convergence topology.^[22] A subbase for this topology consists of sets of the form f^{-1}((a, \infty)) for f \in X and a \in \mathbb{R}, where f: \mathbb{R}^X \to \mathbb{R} evaluates at the point f \in X.^[22] More generally, for Y^X where Y is a topological space, the subbase is \{S(x, U) \mid x \in X, U \text{ open in } Y\} with S(x, U) = \{g \in Y^X \mid g(x) \in U\}, and finite intersections yield basic open sets specifying behavior on finitely many points.^[22] This topology ensures pointwise convergence of nets or sequences.^[22] Subbases are particularly advantageous for defining topologies on infinite products, as directly taking products of bases from each factor may not yield a base for the desired topology, whereas the subbase approach efficiently generates the product topology via finite intersections, avoiding the need for infinite operations and preserving continuity of coordinate functions even in infinite dimensions.^[20] This contrasts with finer topologies like the box topology, which fail to maintain key properties such as the continuity of certain diagonal functions in infinite products.^[20]