Mathematics.

point set topology

Basis for a Topology

Topology35 minDifficulty7 out of 10

You should know: topological space

Overview

Specifying every single open set of a topology directly is often unwieldy, especially for infinite spaces. A basis is a smaller, more manageable collection of subsets — basis elements — from which the entire topology can be reconstructed as all possible unions of basis elements. For example, the open intervals of ℝ form a basis for the standard topology: instead of listing every open subset of ℝ (an intractable task), you only need to describe the intervals, and every open set is automatically a union of some collection of them. A basis is characterized by two simple conditions: the basis elements must cover the whole space, and the intersection of any two basis elements must itself contain a basis element around every one of its points.

Intuition

Think of a basis as a set of building blocks (like LEGO bricks of many shapes and sizes) that you're allowed to combine by simply taking unions — snapping them together — but never by cutting them apart. To guarantee any combination you build this way still obeys the topology axioms, the blocks need to overlap 'compatibly': wherever two blocks overlap, there must be a smaller block available that fits inside that overlap region around any point there, so unions of blocks compose consistently into a well-defined family of open sets. Open intervals (a,b) in ℝ are the classic building blocks: any open subset of ℝ, however complicated, can be assembled by gluing together enough (possibly infinitely many) such intervals.

Formal Definition

Definition

A collection ℬ of subsets of a set X is a basis if it satisfies:

∀x∈X, ∃B∈B with x∈B\forall x \in X,\ \exists B \in \mathcal{B} \text{ with } x \in B

The basis elements cover all of X

Covering condition
∀B1,B2∈B, ∀x∈B1∩B2, ∃B3∈B:x∈B3⊆B1∩B2\forall B_1, B_2 \in \mathcal{B},\ \forall x \in B_1 \cap B_2,\ \exists B_3 \in \mathcal{B}: x \in B_3 \subseteq B_1 \cap B_2

Around every point in the overlap of two basis elements, there is a basis element fitting inside that overlap

Intersection condition
τB={⋃α∈ABα:Bα∈B}\tau_{\mathcal{B}} = \left\{ \bigcup_{\alpha \in A} B_\alpha : B_\alpha \in \mathcal{B} \right\}

The topology generated by ℬ consists of all possible unions (including the empty union, ∅) of basis elements

Topology generated by a basis

Notation

NotationMeaning
B\mathcal{B}A basis: a collection of subsets of X satisfying the covering and intersection conditions
τB\tau_\mathcal{B}The topology generated by the basis ℬ

Properties

Basis for standard topology on ℝ

B={(a,b):a<b}\mathcal{B} = \{(a,b) : a < b\}

Example: Open intervals form a basis; every open subset of ℝ is a union of open intervals.

Basis for metric topology

B={B(x,ε):x∈X,ε>0}\mathcal{B} = \{ B(x,\varepsilon) : x \in X, \varepsilon > 0 \}

Example: Open balls always form a basis for the topology induced by a metric.

Basis criterion for topologies

Given a topology τ, B⊆τ is a basis for τ  ⟺  every open set of τ is a union of members of B.\text{Given a topology } \tau, \ \mathcal{B} \subseteq \tau \text{ is a basis for } \tau \iff \text{every open set of } \tau \text{ is a union of members of } \mathcal{B}.

Comparing topologies via bases

τB1⊆τB2  ⟺  ∀B1∈B1, ∀x∈B1, ∃B2∈B2:x∈B2⊆B1\tau_{\mathcal{B}_1} \subseteq \tau_{\mathcal{B}_2} \iff \forall B_1 \in \mathcal{B}_1,\ \forall x \in B_1,\ \exists B_2 \in \mathcal{B}_2: x \in B_2 \subseteq B_1

Worked Examples

  1. Covering: any real x lies in some interval with rational endpoints, e.g. (x-1,x+1) can be adjusted to nearby rationals surrounding x.

    ∀x∈R,∃a,b∈Q:x∈(a,b)\forall x \in \mathbb{R}, \exists a,b \in \mathbb{Q}: x \in (a,b)

  2. Intersection: the intersection of two rational-endpoint intervals is again an interval (possibly empty), and if x lies in it, a smaller rational-endpoint interval around x fits inside by density of ℚ.

    (a1,b1)∩(a2,b2)=(max⁡(a1,a2),min⁡(b1,b2))(a_1,b_1) \cap (a_2,b_2) = (\max(a_1,a_2), \min(b_1,b_2))

Answer: Yes — this countable collection is a basis, generating the same standard topology on ℝ as all open intervals do.

Practice Problems

Difficulty 5/10

A collection ℬ of subsets of X is a basis if:

Difficulty 6/10

Does ℬ = {{1,2}, {2,3}} form a basis for a topology on X = {1,2,3}?

Difficulty 7/10

Prove that the topology generated by a basis ℬ (all unions of basis elements) actually satisfies the three topology axioms.

Quiz

The topology generated by a basis ℬ consists of:
Why must a basis satisfy the intersection condition (finding a smaller basis element inside overlaps)?

Summary

  • A basis ℬ is a collection of subsets covering X, where intersections of basis elements contain a smaller basis element around each of their points.
  • The topology generated by ℬ consists of all possible unions of basis elements — this is how the standard topology on ℝ is generated from open intervals.
  • Different bases can generate the same topology (e.g. rational-endpoint intervals vs. all open intervals), while different basis structures (e.g. half-open intervals) can generate strictly different, finer topologies.

References

  1. WebsiteWikipedia — Base (topology)