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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