Notes on General Topology Preliminary: Set Theory 1
My notes about Topology.
1. Binary Operation:
Normally, we won't use the term "binary operation" in our daily conversations, but the idea of binary operation is everywhere. For instance, we've learned one of the binary operation called addition like $1+2=3$ and another one called multiplication like $2\times\frac{1}{2}=1$ in our math class. I think you've already noticed that any binary operations takes two inputs elements in a set and gives one output that is also in the same set, and that's indeed how we define it.
Def. A Binary operation on $A$ is $f:A\times A\rightarrow A$.
(Since we've not defined the Cartesian product yet, you can regard "$A\times A$" as a pair of elements of $A$. ) A binary operation is just like a function, but mathematicians use a different notation to denote the value. Usually the output of $(a,b)$ is $f(a,b)$, but in this case, there is a convention to use $afb$ or some other symbols like "$+,-,*,\times$". At first, I was confused about this convention because math really hates abusive notations, but later when I started self-studying the basics of algebraic topology, I realized that these notations actually simplifies the equations because without $*$ the composite of three loops in a path-connected space would be $f(f(\alpha,\beta),\gamma)$ (note that $f$ needs to be specified) instead of $\alpha*\beta*\gamma$.
Besides what we just discussed, the binary operations ($+,\times$) on $\mathbb{R}$ (or $\mathbb{Z}$) bring us a lot of extra algebraic properties like associativity, commutativity; and some order properties. These things motivates the next idea about induction.
2. Induction:
Here, the word "induction", although, is a method of defining sets, it's similar to a technique that we frequently use: mathematical induction.
Def. A set $X\subset\mathbb{R}$ is inductive if $1\in X$ and $x+1\in X$ where $x\in X$.
Let's investigate this definition more! It provides us an order relation because $xnonempty
subset $A$, then $\min_{a_i\in A}\{a_1,a_2,...\}$ exists, otherwise (empty subset) then there's no elements in $\varnothing$. To prove this inductive property, we need mathematical induction! (Interesting isn't it?)3. Cartesian Products:
Before reading this section of his book, my knowledge about the Cartesian product is really limited since my text mainly discuss these things in metric spaces with dimensions less than 4. General topology extends the Cartesian product to arbitrary topological spaces.
Def. The Cartesian product of an indexed family of sets $\{S_i\}_{i\in I}$ is $$\prod_{i=1}^nS_i$$ which can be regarded as the set of $n$-tuples with each coordinate comes from a unique $S_j$ s.t. $j\in I$.
I personally prefer to regard the product as a space with $n$ axes that are perpendicular to each other. (Obviously it's impossible in 3-dimensional space if $dim(\prod_{i=1}^nS_i)>3$) The reason for it is because I think this relates pure algebraic tuples with a geometric realization somehow which is very useful while dealing with the classification of manifolds in my opinion (my personal idea). Maybe someone with a very nice intuition in algebra could construct every map without any graph......
(Note that the product notation $\times$ takes two sets of a indexed family of sets and output a product space that need not be in the family of sets, therefore it's not a binary operation.)
The existence of Cartesian products allows us to extend sequences to higher dimensional by $$s:\mathbb{Z}_+\rightarrow\prod_{i=1}^nS_i$$
The map $s$ actually combines several sequences in each factor space to a sequence of tuples in their product space.
Now, if I replace the RHS of the arrow by $\{A_i\}$, which is a family of sets, and set $s$ to be bijective, then the this map becomes an indexing map that gives us an infinite sequence of spaces, and if we take the cartesian product of $A_i$, then we get the so-called infinite dimensional space that contains the set of $\omega$-tuples. Remember this space because it will make a huge difference after defining the product topology which I'll be writing.
Supplementary Part: An Interesting Fact about Functions (in my opinion):
Here is a knowledge claim: A dog has 4 legs, a cat has 4 legs, therefore, a cat is a dog. This is absolutely ridiculous! (as ridiculous as banana putting...) But I thought about the reason behind this ridiculous statement and I found that it can be explained using some basic knowledge in this topology book.
Let $D$ be the set of all dogs, $C$ the set of all cats, and $F$ the set of all animals that walk with 4 legs, then define two maps $f:D\to F$ s.t. for $d\in D$, $f(d)$ exists if and only if $d$ has four legs and $f(d)=f(e)$ iff $d=e$, similarly, define $g:C\to F$.
It is quite obvious that $f$ and $g$ are injections which means they're one-to-one when restrict their range, but some elements in $F$ have no inverse image (pre-image). See the following figure:
Let $h:D\cup C\to F$ s.t. $h|_D=f$ and $h|_C=g$. Since $f$ and $g$ are both left invertible, $f^{-1}(h(D\cap C))=g^{-1}(h(D\cap C))=D\cap C$. Now, we all agree that $D\cap C=\varnothing$ which implies that the previous statement about dogs and cats is wrong.
