# Quick Revision: Basis and Subbasis (Part 1.2)

## A Better Concept for Basis

A X be a set. Then a collection of subsets of X $\mathcal{B}$ is (almost) called a basis for a topology on X if

1. Union of elements in the collection is the whole of X.
2. Let $B_1,B_2\in\mathcal{B}$. For every $x\in B_1\cap B_2$, there exists a $B_3\in\mathcal{B}$, such that $x\in B\subset B_1\cap B_2$.

The topology generated by this basis is given by all possible union of elements of $\mathcal{B}$.

Note that the second condition, namely, “Let $B_1,B_2\in\mathcal{B}$. For every $x\in B_1\cap B_2$, there exists a $B_3\in\mathcal{B}$, such that $x\in B\subset B_1\cap B_2$” is a more “general” case of the second condition we considered in previous page:

1. Suppose $B_1\cap B_2=\emptyset$, then there is no need to find a $B_3\in\mathcal{B}$ as there is no $x\in B_1\cap B_2$
2. Suppose $B_1\cap B_2\in\mathcal{B}$. Then we can choose $B_3=B_1\cap B_2$ itself!

In other words, it got the second condition in the definition of basis for a topology we discussed in previous page covered! But what does it offer extra (other than making the second condition harder to understand)? Let’s revisit our previous Example 3!

### Revisiting Example 3

Consider the set X = {1, 2, 3, 4, 5}. Let $\mathcal{B}_{X_3} = \{\{1\},\{2\},\{1,2,3,4\}\},\{1,2,5\}\}$

Note that we concluded that it doesn’t satisfy the 2nd condition in previous definition, but let’s look at the new definition.

For $x\in\{1,2,3,4\}\cap\{1,2,5\}\}=\{1,2\}$, for both 1 and 2, we have $\{x\}\in\mathcal{B}_{X_3}$, and hence, that issue is sorted!

In fact, you should have already verified that it generates a topology on X.

This is one of numerous example we can find for basis! Try finding some yourself!

### Example 4

Consider the set of all real numbers. Then the collection of all open inetrvals, $\mathcal{B}_{R_s}=\{(a,b):a,b\in R\}$ is a basis on R and generates a topology on R (with open intervals and their unions as open sets – sounds familiar?). This topology is called the Standard Topology on R and is denoted by $\mathbb{R}$.

### Example 5

Consider the set of all real numbers. Then the collection of all half-open inetrvals, $\mathcal{B}_{R_s}=\{[a,b):a,b\in R\}$ is a basis on R and generates a topology on R, called the lower limit topology. It is denoted by $\mathbb{R}_l$.

Now, definition of basis looks a lot complicated! Is there any way to give an easier “collection” that can generate a topology on X? Turns out yes! But it comes with a trade off!

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