@JonasTeuwen I just upgraded from 1.5Mbps, and I was usually only getting around 800kbps because I was so far from the DSL main office. Now I am on cable modem.
@JonasTeuwen My old ISP did that, but they wouldn't give me the 5 static IPs that I wanted. Then I found out the cable company would, so I changed and got faster, too.
The chapter basically explained the one one correspondence between all topological spaces $(X,\mathfrak I)$ and all nbhd spaces $(X,\mathfrak N)$, and how we can get one from the other.
@PeterTamaroff I hope you've got some good examples to work with. You should at least understand metric spaces pretty well before plowing into topology.
@PeterTamaroff sure, every subset of $\mathbb{R}$ inherits a topology from $\mathbb{R}$ (but you'll get there). And yes, the neighborhoods of $0$ give you exactly the tails you described before.
so I'm perusing some rep theory notes. it takes things like maximal chains and Zorn's completely for granted, but then it says Here i is a complex number such that i^2 = −1 and pi denotes the area of a circle of radius one). strange how views on which topics can be taken for granted and which can't differs between people.
@PeterTamaroff Hm. I learned topology from a German book that was never translated (Boto von Querenburg) and the other books I read are a bit too advanced to recommend. People tend to like Munkres, though.
@anon I think that (at a different level) this is a nice idea. From experience, many people are told, "Oh, well, you're going to do algebraic something-or-other, so make sure to learn algebra well." So you do that, and then you start reading a paper on representation theory and go, "Whoaa look at all this analysis; I have been tricked."
Every time an author reminds me of basic complex/Fourier/functional analysis, I am wildly thankful.
But I can remember the definition of $i$. That's a weird one.
It's very opposite with me. I'd never studied any modern (abstract) algebra until a couple years ago, it was all just analysis. (That was all I was ever offered!)
In my image $f$ maps $A\to B$. The red ovals designate preimages aka inverse images of the elements of $B$ under $b$. We can find a choice function $g:B\to A$ that maps $b$ somewhere into $f^{-1}(b)$. Obviously applying $g$ and then $f$ will preserve all of $B$; it is the identity. Not so in reverse; applying $f$ and then $g$ will map each inverse image to the particular representative of it the choice function chose.
the axiom of choice is subtle....to have a "choice function" it's not enough to have the set on which it is defined...you must also specify the method of choosing
they're letting some people have access to their file collections I heard, but in the process they will be scrutinized so this will only work for a select subset of the 25,000TB.
when people are first exposed to functions, they are of course introduced to things like the domain and image (often called, in my opinon a poor name, range).
@MarianoSuárez-Alvarez I'm really not trying to against the tide here, but my book says "The equivalence relation that is appropriate to topological spaces is called homeomorphism."
