Those of you have heard about topology often hear that ” everything is made of rubbery stuff ” or “you can squish or stretch any shape but never tear or add “, well I used to wonder how all of this is math!
Today we will try to make sense of one the main topics topology, which is homeomorphism. It sounds like a peculiar thing but actually a very basic concept in all of mathematics. Homeomorphism is nothing more than a function( a pair of functions actually!),but it’s not just any kind of function, it’s continuous functions between topological spaces.in the rest of the article we will take step by step to understand
What is Function?
This point I want to shoutout as Thomas Garrity ” Function describe the world“. One can think almost everything in the light of functions, its kind of ‘Mathematical thinking 101’. If you can boil down your problem into a problem of function then there is a good chance you can use some of the mathematical gadgets that mathematicians have already been developed. All of the different fields of mathematics are roughly the study of different type of functions
A function or map is rules that attach every elements of a set to only one element of another set (sets are just well defined collection of objects).we write this as. ƒ : A → B
here A set is called domain and B set is called co-domain, we can also reverse the arrow and if it’s also turn out to be a function than we call it inverse function of f or
f– -1 : A ← B
and there are two properties that give us the necessary criteria for a function to have a successful reverse function or inverse function
these are called one-one and onto
these are also very simple concepts, think about the function f(x) = x^2 is doing, its assigning f(1) = f(-1) = 1 , so if we ask the what elements from the domain is map to 1 we don’t get just one answer we get the pair (1,-1), but remember the function can’t have more than one output against one input ! this why this square function is not one-one thus we can’t successfully reverse it . A function is called one-one if each elements of the domain is mapped to one and only elements in the codomain
as for onto think of this set A = { ( p , q ) : where p ,q are prime numbers besides 2 } and a define function F( p , q ) = p + q .
the codomain is actually the set of even numbers( think about it ! )
A function is called onto if elements of the codomain is the output of some input from the domain, so this function F : A → {set of even natural number besides 2 and 4 } would be onto if every even natural number is the sum of two prime numbers. This is precisely the infamous Goldbach conjecture !! . Proving this F is an onto function is one way to solve this near impossible problem.
If a function is both one-one and onto then we call it a bijection and there exists a way to reverse it unambiguously.
now notice two bijective functions as examples from one of my favourite book
Both of these are reversible function, but they have a very striking difference ! can you tell what it is ?
WHAT IS CONTINUOUS FUNCTION ?
You can think of these Continuous functions are those curves you can draw on a paper without lifting the pen, but this can’t be good definition because it depends on a physical construction, other hand not every type of functions can be drawn on a simple x-y coordinate.
so we need general definition for continuity, for this very reason the field of topology was born, and mathematics is filled with plethora amount of continuous functions, from Zeta function to Hopf fibration
all are deep down continuous functions.
This continuity property is where all the squishy things come into play. Stretching, smudging , squishing are all different metaphors for continuous deformation.
Homeomorphism at last . . .
So a continuous functions not just pick a point move it , it carries a patch of interval containing that point and move it near the output point and it doesn’t end here , it carries infinitely many patches which are arbitrary small but still contain that point and place right near the output point. and it does this infinite fabrication for every points!!(this is beyond inception) that’s what makes continuous functions so robust. It is intricate amount infinite patching infinite amount of times. Truly an embroidery of God.
and homeomorphism is nothing but continuous function between shapes which is also reversable.