make sure to mention that the fix is by Grok for maximum lols

Wait I know this story, I think it was called The Last of Us

what about Andrew Tate bison?

Hey bro, I am not a beggar bro. I am just carrying this termite to the hospital. Can you give some food for the road bro, promise I am not a beggar.

Ah that makes sense, regular definition of basis is not much of use in infinite dimension anyways as far as I recall. Wonder if differentiability is required for what you said since polynomials on compact domains (probably required for uniform convergence or sth) would also work for cont functions I think.

So I call an infinite dimensional vector space of countable/uncountable dimensions if it has a countable and uncountable basis. What is the analytical definition? Or do you mean basis in the sense of topology?

Doesn’t BCT imply that infinite dimensional Banach spaces cannot have a countable basis

brain dump?

because that 300000 people fired will allow them to have 300 high ranking CEOs

Look it is so simple, it just acts on an uncountably infinite dimensional vector space of differentiable functions.

Can he jump from a branch?

No he can’t, he is an ant

Look out, he is a Spider-Ant!

Asking for a friend? For research? No?

Ok, go ahead please.

Ultima Online! I stared at that art for days!

Imagine if he had to apply for funding

“these waves have the potential to transform how we communicate and will likely find world wide usage”

He would actually be right unlike all the other funding applications which are largely oversold.

If solid torus yes, if just the regular torus (surface of the solid torus) no. CD is homotopic to a circle and so is a solid torus.

Even if it has thickness still homotopic to a circle. For instance a band with thickness is homotopic to a circle, you can retract along the radius to arrive at a circle that is inside the band. Similarly a plane, or a slab with thickness are all homotopic to a point.

Note that all of these are proved by using collections of transformations from the space to itself (not necessarily from the space to all of itself though, if it maps the space to a subset of it that is fine). So if you want to say something like “but you can also shrink a circle to eventually reach a point but it is not homotopic to a point” that won’t work because you are imagining transformation that maps a circle not into itself but to a smaller one.

ps: the actual definition of homotopy equivalence between “objects” is slightly more involved but intuitively it boils down to this when you imagine one space as a subset of the other and try to see if they are homotopy equivalent.

What you said is stronger than being homotopic. homotopic is weaker, for instance a line is homotopic to a point, By taking the straw (even if it has thickness) and just shrinking it along its longer axis you eventually arrive at a circle. If it has thickness you will arrive at a band and then you can also retract radially to arrive at a circle.

that bird is doing its best and is very proud of its achievements, leave it alone OK?

a torus is not homotopic to a straw though unless you take the straw and glue it at its ends. a straw is homotopic to a circle, a torus is homotopic to product of two circles, Baldur’s gate is homotopic to a disk which is homotopic to a point unless we are talking about the game storage medium which used to be a CD which is also homotopic to a circle

HOW DARE YOU ASK FOR COMFORT IN YOUR ONLY ONE LIFE???!!