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>> No.11449289 [View]
File: 217 KB, 800x1118, __chen_touhou_drawn_by_poronegi__13b16d6e663e3c1d034d7e0729692a57.jpg [View same] [iqdb] [saucenao] [google]
11449289

We have the set [math]S[/math] of maps [math]\mathbb{Z} \rightarrow \mathbb{Z}_{10}[/math] which are bounded below, that is, for every map [math]f[/math] there is some [math]N[/math] such that [math]n<N[/math] implies [math]f(n)=0[/math].
These are represent "formal strings which look like real numbers". So for example, the element [math]f(0)=0[/math], [math]f(n)=9[/math] if [math]n>0[/math] would be written [math]0.99999 \cdots [/math], [math]\pi[/math] would be [math]g(0)=3[/math], [math]g(1)=1[/math], etc for the n-th digit, and the lower bound on [math]f[/math] really just ensures the integer part is finite.
Anyhow, I want to know if you lads know or can come up with any "no go" theorems for group structures on [math]S[/math], in particular showing that we can't define the real numbers as the lexicographically ordered elements of [math]S[/math] and expect reasonable properties.

>> No.11378288 [View]
File: 217 KB, 800x1118, __chen_touhou_drawn_by_poronegi__13b16d6e663e3c1d034d7e0729692a57.jpg [View same] [iqdb] [saucenao] [google]
11378288

Consider [math]\{ 0, 1\}[/math]. If you place the trivial topology, all bijections are continuous, trivially. If you place the discrete topology, you get the same.
But with the Sierpinski topology, you have that the identity is continuous, but the "twist" bijection isn't.

So I was wondering what topology on a finite set has the most discontinuous bijections.
The solution is as follows:
For [math]\mathbb{Z} _n[/math], the open sets are [math]\mathcal{O}_j = \{ i \in \mathbb{Z}_n | i \leq j \}[/math] as [math]j[/math] runs through [math]\mathbb{Z} _n[/math].
And then the only continuous bijection is the identity (because it needs to send 0 to itself, and if it sends 0 to itself it also needs to send one to itself, and so on), so it maximizes a fortiori.
I was wondering if it still maximized the non-continuous maps if we lifted the restriction on bijections. Any ideas?

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