# Topological Spaces

Remember? Now, take a moment to memorize this definition!

Definition 1: A topology on a set X is a collection 𝒯  of subsets of X having the following properties:

1. ∅, X ∈ 𝒯
2. The union of the elements of any subcollection of is in 𝒯
3. The intersection of the elements of any finite subcollection of is in 𝒯

A set X for which a topology has been specified is called a topological space, denoted by (X, 𝒯)

Definition 2: Let X be a topological space. A subset U of X is called an open set in X if U ∈ 𝒯.

Well, some of you might be thinking – maths is all about memorization right? Never! It’s just that this definition is going to be used over and over again as we proceed. It’s hard to search for this definition each time we have to use it right? That’s why we will try to memorize some basic definitions!

A metric which is attached to a set to form a metric space is…
A function
True that! We defined it as a function with some extra properties!
A set
Oh no! A metric is a function. But you may argue that a function is in turn a set – but closest answer here is “A function”!
A Cartesian product
Oh no! A metric is a function. A function is a subset of Cartesian product of two sets – but the closest answer here is “A function”!
You know that in Algebra, a set along with an operation say, *, denoted by (G, *) is called a group. Then
G and * are functions
Oh no! Try other options!
G is a set and * is a function
True that! G is a set, while an operation, if you recall is a function.
G and * are sets
Oh no! Try other options!

A maths nerd will know that a function is also defined as a subset of a special set called “Cartesian Product” – but it’s so widely used as “sets” that we consider it a basic building block just like sets. Well, what does all these have to do with Topological Spaces, one may ask:

Let be a topology on some set X. Then is
A function
Oh no! Try other options!
A set
True that! is a collection of subsets of X – or in other words, is a set itself!
This is an absurd option since I couldn’t find a fourth.
Oh no! Try other options!
It’s a national secret!
It’s set! Check that option for more details!

Interesting, is it not? While all popular mathematical objects paired themselves with a function to give them a meaning, a Topological Space stands out – it pairs with another set!