Quick Revision: Basis and Subbasis (Part 1.2)

Some Simple Quizzes

Consider X = {1, 2, 3} and the collection . What do you think is ?
, the empty set
That looks wrong! Try picking other options!
X
It’s the whole space X! {1} U {2, 3} = {1, 2, 3} = X
A proper subset of X
That looks wrong! Try picking other options!
It’s a national secret.
It’s the whole space X! {1} U {2, 3} = {1, 2, 3} = X
What all observations have we made about the special collection of subsets of X, so far? (More than one option is correct)
Union of elements in is the whole space X.
True! We noticed it in the last quiz! Do try other options!
All possible unions of sets in generates a topology on X
True again! We found the collection of all possible unions in the last page! Do try the other options!
All possible intersection of elements in is emptyset.
True! In this particular example, possible intersections in $\mathcal{B}$ is empty. Do try other options!
It’s a national secret.
All options are right, click them to know their reasoning!

Do note the way we played with the words there! “Union of all elements” vs. “All possible unions”! By saying union of all elements, we mean taking union of all elements of the collection. With all possible unions, we are talking about a collection of subsets of X, formed by taking union of some sets in the collection (as we did in the previous page to find the topology generated by it).

Consider X = {1, 2, 3} and . Do the same trick we used above. What conclusions can we derive? (More than one options are true!)
Union of elements in is the whole space X.
That looks wrong! Try picking other options!
All possible unions is the PowerSet of the X.
It’s the whole space X! {1} U {2, 3} = {1, 2, 3} = X
Considering all possible unions gives us the discrete topology on X.
That looks wrong! Try picking other options!
It’s a national secret.
It’s the whole space X! {1} U {2, 3} = {1, 2, 3} = X
Well, let’s consider another example! X = {a, b, c, d}, . Which of the following is FALSE for this collection?
Union of elements in is the whole space X.
Oh no! This is true! Verify it yourself!
All possible unions of sets in generates a topology on X
Oh no! This option is also true! Unions gives a topology,
All possible intersection of elements in is emptyset.
This is wrong! Intersection is not empty, but interestingly, intersection itself is contained in this collection!
It’s a national secret.
All except one option is true! Find the false option!

Alright! That’s enough examples! What happens if any of these properties are not met?

Consider X = {1,2,3} and . Will collection of all possible unions form a topology on X?
Yes!
Oh no! {a} is not a subset of X!
Uhm! No! Never
Well! That’s correct! You spotted the villain here, {a}, I hope!
It’s a national secret.
There’s a thief in the colection of subsets of X we are considering, can you spot it?
Consider X = {1,2,3} and . Will collection of all possible unions form a topology on X?
Yes!
Oh no! Does the collection of all unions contain X itself?
Uhm! No! Never
True! Collection of all possible unions does not contain X!
It’s a national secret.
Collection of all possible unions does not contain X! Hence, it cannot form a topology!

Interesting! The property that previous examples held on to, that the union of sets in our collection \mathcal{B} must be equal to X seems to be important!

Consider X = {1,2,3} and . Will collection of all possible unions form a topology on X?
Yes!
Oh no! Again, {1,2} and {2,3} will be there in X, but what about the intersection of {1,2} and {2,3}?
Uhm! No! Never
You’re right! You can choose two sets such that their intersection is {2}. But will {2} be contained in the colleciton of all possible unions?
It’s a national secret.
Try other two options for explanation!

Where we failed in the above quiz is with the intersection criteria we observed earlier. That seems to be important too!

I think we have some idea about what we heading towards. One final detail – what should we name our newborn baby? 😂

Since we took inspiration to search for a way to “compress” data from linear algebra do you think it is a justice to call this new concept a “Basis”?
Yes, yes, yes!
Almost correct! Any improvements you can see among options?
Uhm… no!
Well, like it or not, mathematicians chose to go with this name! 🙂
Maybe more specific, like “Basis for a Topology on set X”?
That looks more likely!
It’s a national secret.
There’s one “almost correct choice” and an “exactly awesome choice”! Try other options!

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