Basis & Sub-basis for a Topology

Introduction

In this blog post, I am going to discuss about basis and sub-basis for a topology.
As the name suggests, basis comprises some sort of building blocks out of which, the whole structure might be constructed or obtained. Just like we talk about a basis in linear algebra. We say, a linearly independent subset $\mathcal B$ of a vector space $V$ is a basis of the vector space if every element $v\in V$ can be obtained by a linear combination of elements of $\mathcal B$ . The same kind of notion appears for topological spaces. Then central object of topological spaces is open set, just like the central object of a vector space is a vector.

What is a Basis?

Suppose, $(X,\tau)$ is a topological space. Recall that the elements of $\tau$ are called open sets. We look for a smaller collection of open sets $\mathcal B$ such that these can generate the other open sets by some elementary operation, namely taking arbitrary union, just like for vector space, the elementary operations on basis which give rise to the whole vector space, are scalar multiplication and addition of basis vectors.
If $\mathcal B$ is a subset of $\tau$ such that:

$U\in\tau\implies U=\bigcup\limits_{i\in I}B_{i},\text{\space for some\space}\{B_i:i\in I\}\subset\mathcal B$
i.e. any open set is a union of some elements of $\mathcal B$ , then we say that $\mathcal B$ is a basis for the topology $\tau$ . The above is equivalent to saying that:

$U\in\tau,x\in U\implies\exists B\in\mathcal B\text{\space such that\space}x\in B\subset U$
Thus, basis elements serve as building blocks for a given topology.

One should notice that, given $\mathcal B$ is a basis for the topology $\tau$ on $X$ , we can deduce the following:

$\bigcup\limits_{B\in\mathcal B}B=X$
$B,B'\in\mathcal B\text{\space and\space}x\in B\cap B'\implies\exists B''\in\mathcal B\text{\space with\space}x\in B''\subset B\cap B'$ .

A natural question to occur is the following: Given a set $X$ , and a collection of subsets of $X$ namely, $\mathcal B\subset\mathcal P(X)$ satisfying the above two properties, is it true that $\mathcal B$ is a basis for some topology on $X$ , i.e. does there exist a topology $\tau$ on $X$ , of which $\mathcal B$ is a basis?

It turns out that the answer is – YES!

Characterization for a Basis

A collection of subsets of a set $X$ for which the properties:

$\bigcup\limits_{B\in\mathcal B}B=X$
$B,B'\in\mathcal B\text{\space and\space}x\in B\cap B'\implies\exists B''\in\mathcal B\text{\space with\space}x\in B''\subset B\cap B'$ .
are satisfied, forms a basis for some topology $\tau$ on $X$ , and then it follows that $\tau$ is the collection $\tau=\{U\subset X:U\text{\space is a union of some elements of\space}\mathcal B\}$ which we call as the topology generated by $\mathcal B$ .

What is a Sub-basis?

Having discussed about bases, let us now explore another important notion, which is closely related to bases, namely sub-bases for a topology.

Given a topological space $(X,\tau)$ , if $\mathcal S$ is a collection of open sets is called a sub-basis if after taking all possible finite intersections of elements of $\mathcal S$ , the resulting collection of open sets, turns out to be a basis of the topology.
In other words: $\mathcal S\subset\tau$ is said to be a sub-basis for the topology $\tau$ , if the collection of all possible finite intersections of elements of $\mathcal S$ , i.e. the collection: $\mathcal B=\{A_1\cap A_2\cap\dots\cap A_m:A_i\in\mathcal S(\forall 1\leq i\leq m),m\in\mathbb N\}$ is a basis of the topology $\tau$ .

Characterization for a sub-basis

Naturally, we would like to explore that given a set $X$ , when does a collection of its subsets $\mathcal S\subset\mathcal P(X)$ , become a sub-basis for some topology on $X$ .

The answer is that it can be any $\mathcal S\subset\mathcal P(X)$ provided that its elements cover $X$ , i.e.

$\bigcup\limits_{A\in\mathcal S}A=X\text{\space or equivalently\space}x\in X\implies\exists A\in\mathcal S\text{\space such that\space}x\in A$

Thus, a sub-basis is essentially a collection of open sets, whose finite intersections of elements create a basis, and when we take arbitrary unions of those basis elements, we get a topology. Thus, it turns out that to specify a certain topology, it is enough to just give a sub-basis or a basis of that topology.

Bases & Sub-bases – Some Examples

(1) For discrete topology, all the singletons $\{x\}$ together form a basis for the discrete topology.
(2) For finite complement topology (co-finite topology) on a set $X$ , all the complements of the singletons i.e. all set of the form $X\setminus\{x\}$ make a sub-basis.
(3) For the usual topology on real line $\mathbb R$ , the bounded open intervals $\{(a,b):a,b\in\mathbb R\}$ form a basis, whereas open intervals of the form $(-\infty,b),(a,\infty):a,b\in\mathbb R$ form a sub-basis. Open intervals $(r_n,s_n)$ with rational end points $r_n,s_n\in\mathbb Q$ form a countable basis of the topology on the real line $\mathbb R$ . Such spaces, having a countable basis, are called second countable.
(4) In a metric space $(X,d)$ all the open balls $\{B(x,r):x\in X,r>0\}$ form a basis for the metric topology.
(5) If we take intervals of the form $\{[a,b):a,b\in\mathbb R\}$ , then this forms a basis for some topology on $\mathbb R$ , different from the usual topology. This is called the lower limit topology on $\mathbb R$ . We call the topological space as the Sorgenfrey Line. A sub-basis for the lower limit topology on $\mathbb R$ is the collection $\mathcal S=\{[a,\infty):a\in\mathbb R\}\cup\{(-\infty,b):b\in\mathbb R\}$ .

One can play around trying to build new topologies generated from different subbases. I can assure that you will end up mesmerizing yourself for sure!

Conclusion

Bases and subbases for topologies turn out to be extremely useful in constructing topological spaces as well as categorizing or classifying topological spaces, because they control entirely how a certain topology behaves. We will encounter more stuff involving bases when we discuss the countability axioms. We will also familiarize ourselves with a close cousin of bases, called local bases. Until then, Stay tuned!