Notes on General Topology (Pre): An Interesting Fact
Today, I'm gonna talk about a common thing in everyone's life --- Counting!
Usually, we set the total sum to be $0$ and add $1$ whenever we meet something new in a set of objects. For example, if we want to count the number of apples in a set, we start by declaring the number to be $0$ and start adding numbers.
Q: Can we count one object twice?
NO! We can't count one object twice because it's not helpful for us to record the cardinality of this set. Thus, the first restriction of counting is "one-one correspondence", in other word, bijective.
Q: Can we count using fractions?
Usually NO! In my opinion, fractions contain relations which are unnecessary for counting, because we want the answer for "How many", not "How much... in comparison to...". Therefore, the connection between objects and integers is the reason why we need to count.
Def. A set $X$ can be counted by a bijection $f:X\to C\subset\mathbb{Z}_+$.
This definition gives us a magical result if we scrutinize its details, $\mathbb{Q}$ is countably infinite!
From the definition of rational numbers, we know that any rational can be written as the quotient of two integers. Thus, every rational number can be defined by using two integers! Formally, we could consider this process as a function defined on two variables. Let $(a,b)\in\mathbb{Z}^2$ be an ordered pair of integers, then $f:\mathbb{Z}_+^2\to\mathbb{Q}$ is defined by $(a,b)\mapsto\frac{b}{a}$, which can be visualized using the chart below.
This definition gives us a magical result if we scrutinize its details, $\mathbb{Q}$ is countably infinite!
From the definition of rational numbers, we know that any rational can be written as the quotient of two integers. Thus, every rational number can be defined by using two integers! Formally, we could consider this process as a function defined on two variables. Let $(a,b)\in\mathbb{Z}^2$ be an ordered pair of integers, then $f:\mathbb{Z}_+^2\to\mathbb{Q}$ is defined by $(a,b)\mapsto\frac{b}{a}$, which can be visualized using the chart below.
Q: How to count them?
Well, obviously you can't finish counting row by row because each row contains infinitely many elements. So, we proceed like the following:
By doing this, we see that $f(n,m)=\frac{m}{n}$ where $\text{gcd}(n,m)=1$ is at the intersection of $n$-th row and $m$-th column. Since $\mathbb{Z}_+^2$ is countable, $\mathbb{Q}$ is also countable and countably infinite.
However, $\mathbb{R}$ is uncountable because $[0,1]\subset\mathbb{R}$ is uncountable, which could be proved using a similar list of real numbers.
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