Notes on General Topology: Cont. and Homeo.

The idea of topology is extremely useful in capturing properties of a space! According to what Munkres has noted in his book Topology, lots of mathematicians including Hausdorff had searched for an appropriate definition of topology for almost all topological spaces before 1920s. Finally, they generalized the idea as follows:

Def. A topology on a set $X$ is a collection $\tau$ of subsets having

1) $\varnothing,X\in\tau$.

2) $\bigcup_iA_i\in\tau$ where $A_i\in\tau$

3) $\bigcap_j^nB_j\in\tau$ where $n<\infty$ and $B_j\in\tau$.

At first, I was extremely confused about such an abstract idea, because it doesn't seem to be capable to provide any detailed information about an object directly, like the number of handles of your coffee mug or the number of holes in your bread... But you'll see its power in just a second!

In elementary analysis, we've learned the equivalence of "limit" definition of continuity and the $\epsilon-\delta$ condition of continuity. Now, the idea of topology gives us a much general definition called the "open-set" condition.

Def. Let $X$ and $Y$ be two topological spaces connected by a map $f:X\to Y$. $f$ is continuous if $B$ open $\implies f^{-1}(B)$ open. i.e. the preimage of any open set is open. i.e. the preimage of any closed set is closed.

Note that this has to be distinguished from the concept of open map and closed map because the directions of implication are different!!!

It's quite obvious that $\epsilon-\delta$ condition is equivalent to the open set condition. Since the open-set continuity is a generalization of $\epsilon-\delta$ condition. Hence it is also equivalent to the limit definition and the net definition, which I'll be going through.

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There is a special case of continuous map that we always care about ---- Homeomorphism! which requires $f$ to be bijective and continuous at both direction. i.e. Both $f$ and $f^{-1}$ are continuous.

Let's see what this gives us:

 

                           $f(x)=x^3$                                   $g(x)=x$

Well, in the first picture, if we neglect the geometric properties of $f$, then the image is topologically the same as the $x$-axis, and so does $g$. Therefore, Homeomoprhism is used to distinguish the topological property of two spaces. 

For an counterexample of homeomorphisms, see this:

$f(x)=(\sin(2\pi x),\cos(2\pi x)), x\in[0,1)$

The problem with this circle is that we can't find a continuous function between $[0,1)$ and $S^1$ with a continuous inverse function.

And this:

$S^1\vee S^1\not\cong[0,1)$

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However, the reason why mathematicians are sure that "homeomorphisms" preserves topological properties before we actually define (co)homology groups still remains unknown for me. Is it a magic? coincidence? or a consequence of extremely accurate math intuition?

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.

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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