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:
- ∅, X ∈ 𝒯
- The union of the elements of any subcollection of is in 𝒯
- 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 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:
be a topology on some set X. Then is is a collection of subsets of X – or in other words, is a set itself!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!
Mathematicos 