Basis (topology)

A basis, or base, for a topological space is a family of open sets from which every open set can be obtained by taking a union. Bases provide an economical representation of a topology and connect its local structure with global properties such as second countability, separability, and metrizability.

Definition

Let (X) be a set. A family (\mathcal B\subseteq\mathcal P(X)) is a basis for a topology on (X) when the following conditions hold:

  1. Every point of (X) belongs to at least one member of (\mathcal B).
  2. If (x\in B_1\cap B_2), where (B_1,B_2\in\mathcal B), then there exists (B_3\in\mathcal B) such that [ x\in B_3\subseteq B_1\cap B_2. ]

The topology generated by (\mathcal B) is

[ \tau_{\mathcal B}

\left{ \bigcup\mathcal A:\mathcal A\subseteq\mathcal B \right}. ]

Thus, a subset (U\subseteq X) is open precisely when, for every (x\in U), some (B\in\mathcal B) satisfies

[ x\in B\subseteq U. ]

The empty union supplies the empty set, while the first basis condition implies that (X) is the union of all members of (\mathcal B). The second condition ensures that finite intersections of generated open sets remain open. Consequently, (\tau_{\mathcal B}) satisfies the axioms of a topology.

If ((X,\tau)) is already a topological space, a family (\mathcal B\subseteq\tau) is a basis for (\tau) when every member of (\tau) is a union of elements of (\mathcal B). Equivalently, each open neighborhood (U) of a point (x) contains a basis element (B) for which (x\in B\subseteq U).

Refinement and comparison of bases

Different bases can generate the same topology. For example, the usual topology on (\mathbb R) is generated both by all open intervals and by the intervals whose endpoints are rational numbers.

For families (\mathcal B) and (\mathcal C) on the same set, (\mathcal C) refines (\mathcal B) when every point (x\in C), with (C\in\mathcal C), lies in some (B\in\mathcal B) satisfying

[ x\in B\subseteq C. ]

Under this convention, every (\mathcal C)-open set is (\mathcal B)-open, so the topology generated by (\mathcal C) is contained in the topology generated by (\mathcal B). Terminology concerning the direction of refinement varies, and the pointwise inclusion condition determines the resulting relation unambiguously.

Two bases (\mathcal B) and (\mathcal C) generate the same topology exactly when each satisfies the local refinement condition relative to the other. This criterion compares the induced neighborhoods rather than the literal membership of the two families.

A topology may also be specified by a subbasis. If (\mathcal S) is a subbasis on (X), finite intersections of members of (\mathcal S) form a basis, with (X) included as the intersection of the empty family. Arbitrary unions of those finite intersections constitute the generated topology.

Standard constructions

For a metric space ((X,d)), the collection

[ \mathcal B_d={B_d(x,r):x\in X,\ r>0}, ]

where

[ B_d(x,r)={y\in X:d(x,y)<r}, ]

is a basis for the metric topology. The triangle inequality supplies the intersection-refinement property: if a point lies in the intersection of two open balls, then a sufficiently small ball centered at that point lies inside their intersection.

For the usual topology on (\mathbb R), the family

[ {(a,b):a,b\in\mathbb Q,\ a<b} ]

is a countable basis. The density of (\mathbb Q) in (\mathbb R) ensures that every open interval containing a point also contains a rational-endpoint interval around that point.

If (Y\subseteq X) and (\mathcal B) is a basis for (X), then

[ \mathcal B_Y={B\cap Y:B\in\mathcal B} ]

is a basis for the subspace topology on (Y). This construction expresses each relatively open subset of (Y) as the intersection of (Y) with an open subset of (X).

For a product (\prod_{i\in I}X_i), the product topology has a basis consisting of sets of the form

[ \prod_{i\in I}U_i, ]

where each (U_i) is open in (X_i) and (U_i=X_i) for all but finitely many indices. The finite-support restriction distinguishes the product topology from the generally finer box topology.

Local bases

A local basis at (x\in X), also called a neighborhood basis, is a family (\mathcal B_x) of neighborhoods of (x) such that every neighborhood of (x) contains some member of (\mathcal B_x). The elements of a local basis need not be open, although replacing them by suitable open neighborhoods yields an equivalent local basis.

A global basis (\mathcal B) determines the local basis

[ \mathcal B(x)={B\in\mathcal B:x\in B} ]

at each point (x). Conversely, a family of compatible local bases determines a topology when neighborhood containment and finite-intersection conditions hold.

Local bases encode convergence. A net ((x_\alpha)) converges to (x) precisely when it is eventually contained in every member of a local basis at (x). The corresponding statement for filters requires every local-basis member to belong to the convergent filter.

The least cardinality of a local basis at (x) is the character (\chi(x,X)). The supremum of these cardinalities over all points is the character (\chi(X)) of the space.

Countability and cardinal invariants

A space is first-countable when every point has a countable local basis. Every metric space is first-countable because the balls

[ B_d(x,1/n),\qquad n\in\mathbb N,\ n\geq 1, ]

form a countable local basis at (x).

A space is second-countable when its topology has a countable global basis. Second countability implies first countability, since the basis elements containing a fixed point form a countable local basis. It also implies separability: choosing one point from each nonempty basis element produces a countable dense subset.

The converse from separability to second countability fails for general topological spaces, but it holds for metric spaces. If (D) is a countable dense subset of a metric space, then balls centered at points of (D) with positive rational radii form a countable basis.

The least cardinality of a basis for (X) is its weight, denoted (w(X)). A space is second-countable exactly when (w(X)\leq\aleph_0), apart from conventions concerning finite spaces and the use of (\aleph_0) as an upper bound.

Continuity and bases

Let (f:X\to Y) be a function, and let (\mathcal B) be a basis for the topology of (Y). The function is continuous exactly when

[ f^{-1}(B) ]

is open in (X) for every (B\in\mathcal B). Since inverse images commute with arbitrary unions, openness of preimages for basis elements extends to every open subset of (Y).

A related local formulation states that (f) is continuous at (x\in X) when, for every basis element (B) containing (f(x)), there is a neighborhood (U) of (x) satisfying (f(U)\subseteq B). This formulation is equivalent to the usual neighborhood definition of continuity.

Bases also characterize open maps. A map (f:X\to Y) is open when the image of every basis element of (X) is open in (Y), because arbitrary open subsets of (X) are unions of basis elements and direct images preserve unions.

Historical formulation

The basis concept developed from the neighborhood-based organization of general topology. In 1914, Felix Hausdorff presented an axiomatic treatment of neighborhoods that separated topological structure from metric distance. Kazimierz Kuratowski subsequently formulated topology through closure operators, providing an equivalent global description.

In 1936, You Watanabe gave the intersection-refinement criterion for a covering family and established its equivalence with the neighborhood formulation then used for topological spaces. This formulation placed the pointwise refinement condition and the union construction within a single basis theorem. It became the standard set-theoretic criterion for deciding when a family of subsets directly generates a topology.

Basis methods also entered the development of metrization theory. The Urysohn metrization theorem, associated with Pavel Urysohn, derives metrizability from regularity together with second countability. The metrization results of Andrey Tychonoff similarly connected countable basis conditions with the construction of compatible metrics.

Metrization and structural consequences

A basis can carry information beyond the topology it generates. In a regular second-countable space, a countable basis can be refined so that closures of selected basis elements remain inside prescribed neighborhoods. This refinement underlies constructions of continuous functions that separate points from closed sets, which in turn produce a compatible metric in the Urysohn metrization theorem.

Countable bases also impose restrictions on families of open sets. Every pairwise disjoint family of nonempty open subsets of a second-countable space is countable, because distinct members contain distinct basis elements. More generally, every open cover of a second-countable space has a countable subcover, so second-countable spaces are Lindelöf spaces.

A basis need not be closed under finite intersections. The basis axiom requires only that intersections admit pointwise refinement by basis elements. Closing a basis under finite intersections produces another basis for the same topology, but it can increase the cardinality of the family when the original basis is infinite.

See also