Bases and Generation of Topologies

Part 4, Chapter 4: Point-Set Topology Basics

Learning objectives

Define a basis $\mathcal{B}$ for a topology
Generate a topology by taking arbitrary unions of basis elements
Recognize the standard basis of open intervals ( $\mathbb{R}$ ) and open balls ( $\mathbb{R}^n$ )
Predict that a finer topology arises from a finer basis

Listing every open set of a topology one-by-one is hopeless for infinite spaces. The standard topology on $\mathbb{R}$ already has uncountably many open sets. A basis lets you describe an entire topology using a far smaller collection of "primitive" sets: every open set is a union of basis elements. Bases are to topologies what generating sets are to groups, or what bases are to vector spaces, a compact description of an infinite structure. Almost every topology you will meet in this book or beyond is specified through its basis.

The definition

A basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets of $X$ such that:

(B1) For every $x\in X$ , there exists $B\in\mathcal{B}$ with $x\in B$ .
(B2) If $x\in B_1\cap B_2$ for $B_1,B_2\in\mathcal{B}$ , then there exists $B_3\in\mathcal{B}$ with $x\in B_3\subseteq B_1\cap B_2$ .

The topology generated by $\mathcal{B}$ is $\tau=\{U\subseteq X:U$ is a union of elements of $\mathcal{B}\}$ (with the convention that the empty union is $\emptyset$ ). It really is a topology: (B1) ensures $X\in\tau$ ; arbitrary unions are immediate; finite intersections follow from (B2).

Three textbook examples

Standard topology on $\mathbb{R}$ : $\mathcal{B}=\{(a,b):acountable basis generating the same topology.

Standard topology on $\mathbb{R}^n$ : open balls $B(\mathbf{a},r)$ form a basis. So do open rectangles $(a_1,b_1)\times\cdots\times(a_n,b_n)$ n). They generate the SAME topology, topologies don't care which basis you use, only what they generate.

Discrete topology on any set $X$ : the collection of singletons $\{\{x\}:x\in X\}$ is a basis (B1 trivially; B2 trivially since two singletons either coincide or are disjoint). It generates the discrete topology because every subset is a union of singletons.

Comparing topologies via bases

Topology $\tau_1$ is finer than $\tau_2$ when $\tau_2\subseteq\tau_1$ , $\tau_1$ has more open sets. Equivalently, every set open in $\tau_2$ is also open in $\tau_1$ . The discrete topology is the finest possible on any set; the indiscrete topology is the coarsest. The standard basis $\{(a,b)\}$ generates the standard topology on $\mathbb{R}$ ; the half-open basis $\{[a,b)\}$ generates the strictly finer Sorgenfrey topology, every standard-open set is Sorgenfrey-open, but $[0,1)$ is Sorgenfrey-open and not standard-open.

Use the Venn widget to think about (B2): given two basis elements that overlap, you need a smaller basis element inside the overlap containing your point. In $\mathbb{R}$ , when $x\in(a_1,b_1)\cap(a_2,b_2)$ , you can take $B_3=(\max(a_1,a_2),\min(b_1,b_2))$ , the intersection itself is in the basis. Not every basis has this convenience, in general you just need some basis element inside the overlap.

Where this shows up

Network analysis, neighbourhood bases: Computational topology on graphs uses small neighbourhood bases (ego-networks, k-hops) as "open sets" for local computation. Algorithms like community detection (Louvain, label propagation) exploit the fact that local information from neighbourhood bases reconstructs the global topology.

Molecular biology, conformation space topology: The space of all possible folded configurations of a protein is parameterized by torsion angles. A basis for the "natural" topology consists of small clusters around energy-minimized conformations. Molecular-dynamics simulations explore the topology by sampling these basis elements.

Machine learning, manifold learning: Algorithms like UMAP and t-SNE build a local basis at each data point (k-nearest neighbours) and stitch these local pieces into a global low-dimensional embedding. The topology is implicit in the choice of basis, how local "neighbourhoods" are defined.

Pause and think: The collection of open intervals $(a,b)$ with $a,b\in\mathbb{Z}$ , integer-endpoint intervals, is a basis for the standard topology on $\mathbb{R}$ . Why does it generate the same topology as the basis of ALL open intervals, even though it has fewer elements? (Answer: every open interval $(p,q)$ with real endpoints can be written as a union of integer-endpoint intervals: $(p,q)=\bigcup_{n,m\in\mathbb{Z},\ p

Try it

Predict first: which topology does the basis $\mathcal{B}=\{(a,\infty):a\in\mathbb{R}\}$ generate on $\mathbb{R}$ ? (Answer: the "lower-limit ray" topology, open sets are rays $(a,\infty)$ and their unions, plus $\emptyset$ . This is strictly coarser than the standard topology because bounded intervals are NOT open.)

The half-open intervals $[a,b)$ for $a

Predict: does the collection $\{[a,b]:a\leq b\in\mathbb{R}\}$ , closed intervals, form a basis for a topology on $\mathbb{R}$ ? Check (B2): if $x\in[a_1,b_1]\cap[a_2,b_2]$ , can you find a closed interval $[a_3,b_3]$ with $x\in[a_3,b_3]\subseteq[a_1,b_1]\cap[a_2,b_2]$ ? Yes, take $a_3=b_3=x$ , the singleton $\{x\}=[x,x]$ . So yes, this is a basis, and it generates the discrete topology.

Verify: in $\mathbb{R}^2$ , open rectangles $(a,b)\times(c,d)$ generate the same topology as open balls $B(\mathbf{x},r)$ . (Hint: every rectangle contains a ball around each point, and every ball contains a rectangle around each point.)

Trap: a sub-basis is weaker than a basis. The collection $\{(-\infty,b):b\in\mathbb{R}\}\cup\{(a,\infty):a\in\mathbb{R}\}$ is NOT a basis (the intersection $(-\infty,b)\cap(a,\infty)=(a,b)$ is not in the collection), but it generates the standard topology after taking finite intersections first. This is the difference between a basis (closed under "intersection-locally") and a sub-basis (no such requirement).

A trap to watch for

A collection $\mathcal{B}$ being a basis is a property of $\mathcal{B}$ alone (axioms B1 and B2), but two different bases can generate the SAME topology, topology cares only about what is generated, not the generating set. Conversely, two bases that LOOK similar can generate different topologies: open intervals $(a,b)$ vs. half-open intervals $[a,b)$ generate the standard vs. Sorgenfrey topology respectively. Always check whether a basis manipulation changes the generated topology, visual similarity does not equal topological equivalence.

What you now know

You can verify the basis axioms (B1, B2), use a basis to describe the open sets of a topology, and compare topologies by relating their bases. This wraps up the topological-foundations chapter, you now have the vocabulary (topological space, open/closed sets, interior/closure/boundary, compactness, metric, basis) that the rest of mathematics rests on.

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed. Cambridge UP, ch. 4.

Munkres, J. R. (2000). Topology (2nd ed.). Prentice-Hall, ch. 2 (section 13).

Willard, S. (1970). General Topology. Addison-Wesley, ch. 5.

Kelley, J. L. (1955). General Topology. Van Nostrand, ch. 1.

Hatcher, A. (2002). Algebraic Topology. Cambridge UP, appendix.

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