/sci/ - Science & Math

>TFW 2ND PLACE

>>9642155
It's a long process of me hating myself and living in a hostile environment, but at the same time trying to justify my existence by trying to be good at at least something for once. My parents would like to see me get a PhD, but I don't know if I'm capable of even getting in anywhere. I can try, but I don't know if I can handle my potential failure, and I can just forget the whole thing and live the rest of my life in the shame of giving up and without doing the only thing I've ever been decent at. I guess I appear smart to those who don't know uni math is really like since everyone is telling me I'll make it and other stupid shit like that, but I only see a trail of failures behind me. You seem nice, and I don't want to drag you to the bottom with me.

>>9628285
I've been avoiding all that stuff for nothing... All this time...Thanks. You gave me something to think about.

>>9040785
The K being both a field and that set confused me, np. It does work for sets of 2 elements. You will then get a pair of trivial groups, and this would give a trivial gorilloid, but the gorilloid sturcture would be justified iff there would be a non-trivial example. Sry for being mean.

>>8981158

Moreover, this gives a functor [math]\beta\colon\textbf{Top}\to\textbf{CHaus}[/math]: For any continuous map [math]f\colon X\to Y[/math], then [math]\beta(f)\colon\beta X\to\beta Y[/math] is the unique map such that [math]j\circ f=\beta(f)\circ i[/math], where [math]j\colon Y\to\beta Y[/math] is the associated map of [math]Y[/math]. It follows that:
1) [math]\beta(1_X)\circ i=i\circ 1_X=i[/math], and, by uniqueness, [math]\beta(1_X)=1_{\beta X}[/math].
2) Let [math]f\colon X\to Y, g\colon Y\to Z[/math] be continuous. Then, if the associated maps are [math]i_X, i_Y, i_Z[/math], we have [math]i_Y\circ f=\beta(f)\circ i_X, i_Z\circ g=\beta(g)\circ i_Y[/math], and this gives us [math]i_Z\circ g\circ f=\beta(g)\circ i_Y\circ f=\beta(g)\circ\beta(f)\circ i_X[/math], and again, by uniqueness, [math]\beta(g\circ f)=\beta(g)\circ\beta(f)[/math].
Now, let [math]I\colon\textbf{CHaus}\to\textbf{Top}[/math] be the inclusion functor. Since the maps [math]\beta(f)[/math] are unique, there is a bijection [math]\text{Hom}(\beta X, C)\to\text{Hom}(X, IC)[/math], that is, [math]\beta[/math] is a left adjoint of [math]I[/math]. Adjoints preserve various things, so you should check what kind of stuff is preserved by a left adjoint.

>mfw i hit the character limit

The understanding was lost when Atlantis went underwater. They used relativity for weaponry in the Finno-Korean hyperwar back in the days. We had it all, but we just had to fight eachother, and now our civilization is nothing but a small fragment of what once was.

>>8673026
Oh, I was just a total dimwit. Sorry. Dismiss my post.