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:
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:
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?
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