Quick Revision: Basis and Subbasis (Part 1.2)

Subbasis for a Topology on a Set

The trade off where we stopped at in previous page is that with simplicity in definition, comes complexity in calculation. Let’s see how.

Let X be a set. A collection of subsets of X \mathcal{S} is called a subbasis for a topology on X if the union of sets in \mathcal{S} is the whole of X.

That looks familiar! Where did you see it before?
The first condition in the defnintion of basis.
True! Definition of subbasis looks like simply definition of basis with second condition dropped!
The second condition in the definition of basis.
Oh no! Wrong! Check the other option for details!
It’s a national secret.
The first condition in definition of basis is…! Check that option for details!

Interesting! But how to find a topology from it?

Topology generated by a subbasis \mathcal{S} is the collection of all possible union of finite intersection of sets in \mathcal{S}.

Example 1

Consider X=\{1,2,3\}, \mathcal{S}=\{\{1,3\},\{1, 2\}\}.

Clearly, union of all elements in \mathcal{S} = \{1,3\}\cup\{1,2\}=\{1,2,3\}=X.

That’s just one of many examples, try generating more yourself!

We’ve had plenty enough examples – but are they enough? The next post (when posted) will be about many more interesting examples of topological spaces! Watch out Facebook Page/LinkedIn page for more details!

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