Topological Spaces

Hope you remember interesting properties of open sets we noticed earlier. Forget everything else, and let’s focus on those interesting properties alone.

Imagine collecting all the open sets in $\mathbb{R}$! That one big collection will interestingly have these properties as we observed earlier:

1. ∅, $\mathbb{R}$ will be in that set, as we noticed that they are open. In fact, that will be the only pair of complements which are both open in this collection as we observed earlier.
2. Take union of some elements in that set! We know that union of open sets will again be open, hence, it will be contained in our interesting set.
3. That goes for finite intersectin of elements again – finite intersection of elements in this collection of open sets will again be in this collection!

Interesting, isn’t it? This collection seems to be “self-contained” or has a “closure” with respect to the three basic set operations! Can we find other sets like this? Let’s experiment.

Consider the collection of subsets of given by . Does this , which is a collection of subsets of satisfy the above properties?
Yes!
Oh no! Check the other option for the reason!
No!
That’s correct. It doesn’t satisfy all the conditions listed above! If we take , is the union in ?

Our first experiment seems to have failed! Do not be sad, or give up yet! What if we add that missing set to our collection?

Consider the collection of subsets of given by . Does this , which is a collection of subsets of satisfy the above properties?
Yes!
True that! First property is directly evident. Take all possible union of elements in – it’s again in ! Same goes with intersection!
No!
Oh no! Try the other option for the reason!

That’s curious! There seems to be other collections of subsets of $\mathbb{R}$ which satisfies this condition! But such collection – should it always be a collection of subsets of real numbers?

Let’s see:

Let X = {a, b, c}. Consider the collection . Then
are in
That’s correct! Don’t forget to check the other conditions!!
All possible union of elements in are in
That’s correct, too! ,…. (don’t forget to check all possible combinations!)
All possible (finite) intersection of elements of are in
That’s correct, too! , …. (don’t forget to check all possible combinations!)

Well, this shows that we can find a collection that satisfies this condition for any given set! Anything with self-containing properties like this might hide some beautiful properties, so mathematicians decided to explore this “mysterious” object further – which lead to the discovery of the fantastic hero of our story here – Topological Spaces!

A quick question before we proceed. Let X be any set. Then, will the collection satisfy all three of the above properties?
Yes
True! Take union/intersection of any two sets (only possible combinations are – they are all contained in !
No