Recent content by Landau

You are focussing on words rather than the meaning of words. I suggest you first try to understand how the facts you state imply that O(n) is not connected. * connectedness is a property of a topological space, not of a group or a set * O(n) is partitioned into the 1 and -1 fibers of the...

Well, it consists of all elements of the form r(x^2+1)+s(x+1) with r,s ring elements. So it cannot contain, e.g., the element x.

You cannot decide whether a given set is a field. You can decide whether a given ring is a field. After all, a field is by definition a special kind of ring, namely a commutative ring in which every nonzero element is invertible. In particular, a field has no zero divisors (i.e. a field is a...

A subnet is not so hard to define, although there are slightly different ways which give the same nice properties: a set is compact iff every net in it has a convergent subnet, a net converges to x iff every subnet converges to x, and the like. I was just watching the last hours of Day 6 of 24...

Yes, that's what I meant as well (larger in the sense of inclusion). I am sorry, that is of course what I meant (I wrote mistakinly 'discrete' instead of 'trivial'). The finer=larger=stronger the topology, the less convergent nets. I think I understand what you are saying. You seem to be...

A topology is uniquely determined by its convergent nets; in general sequences do not suffice. So yes, if you start with a collection sequences with a specified limits, and demand that these are all the convergent nets, then you get a topology whose closed sets are the sequentially closed...

No it doesn't. <x,0>=x1x2.

You just said it yourself. View U(n) as subspace of R^{2n^2}. You know the open sets of R^{2n^2}, hence of every subspace of it.

Ah, so by 'induced on [0,1]' you don't mean the subspace topology. Could you define the fine topology for me? Is it the initial topology on X w.r.t. all convex functions X->R?

I think he wants to show that this equivalence relation (or rather, 'being tangent') is chart-independent, i.e. that if two curves are tangent in some chart, then also in any other chart. This follows directly from the chain rule.

What do you mean by 'equivalent' topologies? I am not familiar with the fine topology, but if by equivalent topologies you simply mean 'the same topology' (i.e. the same open set), then it is of course a tautology.

No, the author was correct: he did not mean that Nx \cup Ny is a filter base, but a filter subbase. A filter subbase is a collection subsets, such that every finite intersection is non-empty. Then the collection of those sets which contain such a finite intersection forms a filter, it is the...

You have proven that 1/0 does not obey the usual rules of computing with fractions. This should not be too surprising, and doesn't have much to do with 'infinity'.

I agree with DH. Also see here.

Mod out by [G,G], its commutator subgroup. E.g. see here.