## Bases

Recall that though a subring or ideal of a ring may be rather huge, it often suffices to specify just a few elements which will generate the subring or ideal. Likewise, in a topology, one can specify a few open sets and generate the rest via unions and finite intersections. We’ll expound upon that in what follows.

First, note that when (*X*, *d*) is a metric space, **a subset U of X is open if and only if it is a union of (possibly infinitely many) open balls**. Indeed, since every open ball is open, so is a union of multiple open balls. Conversely, if

*U*is open, each element is contained in some open ball for some ε>0 which depends on

*x*; hence

*U*is the union of all these

*N*(

*x*, ε).

In other words, consider **B** = {*N*(*x*, ε) : *x* in *X*, ε>0}, the collection of all open balls; a subset of *X* is open if and only if it is a union of elements of **B**. Generalising this concept for topological spaces gives us the following definition.

Definition. Abasisfor a topology (X, T) is a collection of open sets, such that every open subset U of X is a union of elements . We shall call an element abasic open set.

**Examples**

- As we saw above, the set
**B**of open balls in a metric space (*X*,*d*) forms a basis of the induced topology. - In particular, the set of open intervals (
*a*,*b*) in**R**forms a basis. - In the discrete topology, the collection of singleton sets {
*x*} forms a basis. In fact, if you take the collection of open balls {*N*(*x*, 1/2)} for the discrete metric, you get the same basis. - Since the metrics
*d*,*d*_{1}and*d*_{∞}on**R**^{n}give rise to the same topology, we can pick the set of open balls under any one metric to produce a basis. - Exercise: find all possible bases of
**N**under the right-order topology. [ Answer: there’s no non-trivial basis, i.e. any basis must include all open sets. ]

Our next question is: suppose is some collection of subsets of *X*. Is it always the basis of some topology?

If it were, then clearly the resulting topology would be:

Let’s check if *T* satisfies the axioms of a topology.

*Empty set and X*: the empty set can be written as a union of an empty collection of , so no problem there; for*X*, we need to specify that .*Arbitrary union*: this is obvious, since the union of sets (where each ) is also of the form which lies in*T*.

*Finite intersection*: write and , with each . Since intersection is distributive over union, we get:

For this to be in *T*, a sufficient condition is that for all . On the other hand, this condition is obviously necessary since if , we have . Thus we have proven:

Theorem. A subset is a basis for some topology if and only if:

- the union of all is the whole X; and
- for any , the intersection is a union of elements from B.

## Application: Furstenberg’s Proof of the Infinitude of Primes

While he was an undergraduate, Hillel Furstenberg found an innovative proof that there’re infinitely many prime numbers, via the concept of topology.

Theorem. There are infinitely many prime numbers.

**Proof (Furstenberg)**

Define a topology on set of integers **Z** as follows. For integers *a*, *b* let *V*(*a*, *b*) be the set of all integers *am*+*b* for integer *m*. Then any intersection corresponds to solutions of simultaneous linear congruences , . From elementary number theory, this intersection is either empty or *V*(*e*, *f*) for some integers *e*, *f* (where *e* = lcm(*a*, *c*)). Since *V*(1, 0) = **Z**, {*V*(*a*, *b*)} forms a basis for some topology *T* on **Z**.

Since the complement **Z** – *V*(*a*, *b*) is a union of *V*(*a*, *b’*) for various *b’*, it is open, i.e. *V*(*a*, *b*) is both open and closed. Now, since any integer other than ±1 is divisible by some prime *p*, we have:

If there’re finitely many primes, then the RHS is a union of finitely many closed sets and is hence closed. Thus, {-1, +1} is open, which is ridiculous: since every basic open set *V*(*a*, *b*) is infinite, every open set must be infinite too. ♦

## Subbases

Let’s fix an underlying set *X* and consider various topologies on it. If is a collection of topologies on *X*, one easily checks that so is where a subset of *X* is open in *T* if and only if it’s open in all *T _{i}*. In short,

**an intersection of topologies is still a topology**; so given any subset , one can consider the collection of all topologies containing

*S*(this is a non-empty collection since it includes the discrete topology) and take their intersection.

Definition. Under the above definition, the resulting topology is called thetopology generated by S. One also says that S is asubbasisfor the topology T.

Put in another way, *T* is the “smallest” topology on *X* containing *S*, in the following sense:

- ;
- if
*T’*is any topology containing*S*, then .

[ This is similar to case where an arbitrary subset of a group or ring can be used to generate a subgroup or subring. ]

Theorem. The topology T generated by subbasis S has, as basis,

[ Note: the additional term {*X*} may be superfluous if one interprets *X* as being the intersection of no *W _{i}*‘s. The more terms we intersect, the smaller the set becomes, so if we intersect no terms at all, we get the universal set

*X*.]

**Proof**.

- The intersection of any two elements of
*B*(*S*) still lies in it, so*B*(*S*) is indeed a basis for some topology*T’*. - Since , we have too, since
*T*is the smallest topology containing*S*. - Finally, if , for some , then since , we have as well. Thus, . ♦

In general, bases and subbases can be useful tools in topological proofs since *to verify a property, it often only suffices to do it on basic open sets, or even subbasic open sets*.

Summary. We have:

**Examples**

- On
**R**, the set of intervals of the form (-∞,*b*) and (*a*, ∞) forms a subbasis; indeed, the intersection gives (-∞,*b*) ∩ (*a*, ∞) = (*a*,*b*) or the empty set. - If |
*X*|>2, then the collection of two-element subsets {*a*,*b*} forms a subbasis since if*a*,*b*,*c*are distinct, then {*a*,*b*} ∩ {*b*,*c*} = {*a*}. - For the topology on
**Z**in Furstenberg’s proof, the set of*V*(*a*,*b*) for prime power*a*forms a subbasis by Chinese Remainder Theorem.