Excellent, thanks for the reference.

> I strongly suspect you should be possible to find it in the pdf form available somewhere, if you look hard enough.

I have similar suspicions 🙂

]]>I was talking about this book:

http://www.amazon.com/Seifert-Threlfall-textbook-topology-Mathematics/dp/0126348502#_

I strongly suspect you should be possible to find it in the pdf form available somewhere, if you look hard enough. It seems like a pretty good book, too.

]]>> Yes, you are going to make dozens of amazing mathematical discoveries, all of which will turn out to be already known, before you finally find the one which is really new. But at least you haven’t discovered integration, like this guy did

haha. It is somewhat heartening that this happens to everyone.

> If I remember correctly, before modern axioms of the topological space solidified in their current form, they were working with “neighborhood spaces” which were the exactly same thing, but with a different set of axioms, which was at the same time more ackward and more intuitive than the one we use now.

I’ll have to look into those. A first search didn’t tern up much, but I can probably find Seifert’s book…

> Nitpicking about the last paragraph: the huge difference between Banach and metric spaces is that Banach must be linear. (or vector space, whichever term you prefer.

Fair enough. 🙂

> What I find challanging, is to visualize a topological space…

Yeah, I find the same thing. I’ve kind of given up on visualizing topologies beyond lots of quasi-venn-diagram things.

That said, I just started to reading about algebraic topology, and I’m kind of awed by the power of homology groups to summarize such complicated ideas. It doesn’t deal with visualization but it is… a good substitute for one, I guess? Of course, it doesn’t help with the detailed point set stuff.

]]>That feeling is so-o-o familiar… 😀 Yes, you are going to make dozens of amazing mathematical discoveries, all of which will turn out to be already known, before you finally find the one which is really new. But at least you haven’t discovered integration, like this guy did:

If I remember correctly, before modern axioms of the topological space solidified in their current form, they were working with “neighborhood spaces” which were the exactly same thing, but with a different set of axioms, which was at the same time more ackward and more intuitive than the one we use now. I don’t remember the details right now, but I’ve read about it in the well-known book by Seifert. I am pretty sure you can find it if you don’t already have it.

Nitpicking about the last paragraph: the huge difference between Banach and metric spaces is that Banach must be linear. (or vector space, whichever term you prefer.) In the general metric space it is impossible to define even the concept of a straight line, while in the linear space one doesn’t even need the norm to do that. Even without a norm, linear space is a relatively rich object.

What I find challanging, is to visualize a topological space. Some spaces are difficult to visualize because they are much more complicated than the boring old R^3 we have used to: infinitely dimensional spaces, curved manifolds and the like. But some other spaces are hard to visualize because they are much more poor than ours – take matric space, for example, which doesn’t have angles or straight lines. And topological spaces are poorer still, they make metric spaces almost seem rich.

If a topological space has only finitely many elements, I always picture them as potatoes, and open sets as transparent plastic bags that contain some of those potatoes. Sometimes if you can use homeomorphism: if a space you can’t imagine is homeomorphic to the one you can, you can pretend to be working with the latter. But I don’t know any general way to do it, and that is why [general] topology frightens me.