Quick Revision: Basis and Subbasis (Part 1.2)

Compressing Information

All of us must have heard about ZIP Archives – that helps us compress a file/set of files into a smaller size so that it can be easily attached to emails. Is there any way to represent a topology in terms of a “smaller” collection of sets?

The search for a smaller collection that describes something big – where have we heard it before? You’re right, remember basis in linear algebra?

They helped us represent a very big fields like \mathbb{R}^3 in using 3 elements in it (namely {(1, 0, 0), (0, 1, 0), (0, 0, 1)}) along with \mathbb{R} – remember?

Can we do something similar topology we just defined now? Find a smaller collection, which with some magical spell gives us the entire topology?

Let X = {1, 2, 3}. Consider this collection of subsets of X: \mathcal{B} = \{\{1\}, \{2, 3\}\}.

Let’s try taking all possible unions of sets here.

  1. At first, union of no sets gives us \emptyset.
  2. Singleton unions gives us {1}, {2, 3}.
  3. Well, there are only two sets left – let’s take their union: \{1\}\cup \{2, 3\}=\{1, 2, 3\} = X.

Let’s take the collection of all sets we found using magical union: \{\emptyset, \{1\}, \{2, 3\}\}, X\}

Is a topology on X?
Yes
True that! It statisfies all conditions in the definition of topology!
No
Oh no! Try other options!
It’s a national secret!
Aye aye! The answer is yes, check that option for more details!

Interesting! We could generate a topology which is bigger with a smaller collection of sets. Since union is the magic spell we are using, this topology generated by such a collection will be unique, too.

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