User:Lord Farin/Sandbox

This page exists for me to be able to test out features I am developing. Also, incomplete proofs may appear here.

Feel free to comment.

Over time, stuff may move to User:Lord_Farin/Sandbox/Archive.

Improvement of Sequence of Implications of Connectedness Properties
For brevity, let us introduce the following acronyms:


 * $\mathrm{AC}$: Arc-Connected
 * $\mathrm{UC}$: Ultraconnected
 * $\mathrm{PC}$: Path-Connected
 * $\mathrm{HC}$: Hyperconnected
 * $\mathrm C$:  Connected

Then the following sequence of implications holds:


 * $\begin{xy}

<-3em,3em>*+{\mathrm{AC}} = "AC", <3em,3em>*+{\mathrm{UC}} = "UC", <0em,0em>*+{\mathrm{PC}} = "PC", <6em,0em>*+{\mathrm{HC}} = "HC", <3em,-3em>*+{\mathrm C}  = "C",

"AC";"PC" **@2{-} ?><>(1.2)*@2{>}, "UC";"PC" **@2{-} ?><>(1.2)*@2{>}, "PC";"C" **@2{-} ?><>(1.2)*@2{>}, "HC";"C" **@2{-} ?><>(1.2)*@2{>}, \end{xy}$

What do you think? Now that I have created the moulds, it will be easy to adapt to the other 'Sequences of Implication'. --Lord_Farin (talk) 10:21, 31 August 2012 (UTC) It be noted that it will forever be impossible to endow diagrams (and indeed, any TeX rendered with MathJax) with internal links; sorry. --Lord_Farin (talk) 10:22, 31 August 2012 (UTC)


 * No response? :( --Lord_Farin (talk) 21:31, 7 September 2012 (UTC)


 * It's very nice! I don't know the first thing about what it meant, though :) --GFauxPas (talk) 21:53, 7 September 2012 (UTC)


 * Sorry, only just noticed it. Very nice - one caveat: you need to refer to the key to (a) work out what the codes mean, and (b) to get to the link explaining them. The somewhat clumsier page from which the original of this came does have the full map as one self-contained unit. Might be interesting to put the two presentations up on the same page as alternative renditions. --prime mover (talk) 22:21, 7 September 2012 (UTC)

Group Example: inv x = 1 - x
I think I have it - finally. Reason along with me; denote $\simeq$ for "is bijective/homeomorphic".

We have $\left({0 \,.\,.\, 1}\right) \simeq \left({1 \,.\,.\, \infty}\right)$ via $x \mapsto x^{-1}$.

We have $\left({1 \,.\,.\, \infty}\right) \simeq \left({0 \,.\,.\, \infty}\right)$ via $y \mapsto y - 1$.

We have $\left({0 \,.\,.\, \infty}\right) \simeq \R$ via $z \mapsto \log z$.

Consequently $\left({0 \,.\,.\, 1}\right) \simeq \R$ via $\phi: x \mapsto \displaystyle \log \left({\frac {1 - x} x}\right)$.

Define $\circ = \phi_*+$ as the binary operation induced by $\phi$ and addition $+$ on $\R$, i.e.:


 * $x \circ x' := \phi^{-1} \left({\phi \left({x}\right) + \phi \left({x'}\right)}\right)$

The identity is $\phi^{-1} \left({0}\right)$, and needs to satisfy $\frac{1-x}x=1$; thus is $x = \frac12$.

I couldn't locate a page defining such an induced operation through a bijection $\phi$ but recalled reading about it on some page; it needs to be located, defined and referred to. I'm happy that there is not a flaw in Clark, and that I've been able to come up with the solution. --Lord_Farin (talk) 14:29, 9 October 2012 (UTC)


 * Brilliant! --prime mover (talk) 15:10, 9 October 2012 (UTC)