Lets invent Euler Number!

Lets invent Euler Number!

Why should we study real analysis? Once you are inside the first course of this beautiful subject, you will find hundreds of complicated theorems and formulas floating around the canvas. What makes this journey into rigorous deductive reasoning interesting is: we get...
Proper Metric Spaces can be modeled by Rays!

Proper Metric Spaces can be modeled by Rays!

‘Proper’ is a heavily overloaded term, both in life and in mathematics. It may mean different stuff in different contexts. Thankfully mathematics is far less complicated that life and we can rigorously define properness. Proper Function: A continuous function \( f : X...
Compact Set, Proper Spaces and Annulus

Compact Set, Proper Spaces and Annulus

Euclidean Spaces have a very nice property. In \( \mathbb{R}^n \) (equipped with standard Euclidean metric), every closed and bounded set is a compact set. The converse is also true. Every compact set is closed and bounded). This property is known as Heine Borel...
Why is it interesting to laminate a genus-2 surface?

Why is it interesting to laminate a genus-2 surface?

We are interested to understand the structure of a group G (is it abelian, or a product of abelian groups, or a free group of some kind etc.). In general, this is a (very) hard question. One strategy is to let G act on some well known (topological) space. Often by...