/sci/ - Science & Math

topology

>>9340241

>>9101522
Let [math]x = ab[/math] and denote [math]\phi_n:G \rightarrow G[/math] the n-th power map, then [math] \phi_n(x) = \phi_n(ab) = (ab)^n[/math] but [math]\phi_n[/math] is an homomorphism so [math]\phi_n(ab) = \phi_n(a)\phi_n(b) = a^n b^n[/math] and [math](ab)^n = a^n b^n ~\forall a,b\in G[/math] iff [math]G[/math] is commutative.

>define a set [math]^*\mathbb{R} = \mathbb{R} \cup \{\epsilon, -\epsilon\}[/math] that extends all the first-order operators and axioms (i.e. all axioms except the completeness axiom) as if [math]\epsilon, -\epsilon[/math] were real variables
>define an unsurprising axiom [math]-(\epsilon) = -\epsilon[/math]
>define a function [math]\mathbb{st}(^*x) = \lim_{\epsilon \to 0^+} \lim_{-\epsilon \to 0^-} {}^*x \in \mathbb{R}[/math]
Is this not an easy construction of the theory of the hyperreals? I'm thinking I must be missing something, since it wasn't particularly hard to come up with. This is essentially a formal summary of my rudimentary understanding of hyperreal numbers so bear with me if I've overlooked things. I can perform differentiation and take limits in general successfully and I find it much more intuitive with a better notation.

>>8508182
I neither confirm nor deny that statement. Non-constructivists BTFO.

>>8412608
Have you studied homotopy theory or algebraic topology? They are cool, too. DYEL?

Because I want to understand. To be honest, I don't think I'll ever reach any big results if I even get the chance to become a researcher, and thus it all comes back to me. I want to understand.

And just by looking at the CGI neuron synapse as the videos preview picture, I know the video won't cover stuff I'm into. A theory of everything usually covers only the physical world.

>>8328567
>>8327197

>>8327294
and lastly, my favourite

>>8316112
What do you think?

>>8312708
Atiyah.
>yfw

>>8030689
Incorrect. A real number is the [math]{\it equivalence~class}[/math] of the Cauchy sequences on [math]\mathbb{Q}[/math]. Convergence is just an artifact of the fact that [math]\mathbb{R}[/math] is complete.
>>8030692
>being this retarded
[math]x[/math] is a limit point in the topological space [math]X[/math] if every neighborhood [math]x \in U[/math], [math]U \in \tau(X)[/math] intersects [math]X[/math]. You don't even need a metric for this.