Topology and Geometry §1.6 - §1.17

[20:46] * breeden changes topic to 'Readings in Geometry and 
Topology seminar.  During the seminar, please type ! if you have a 
question and wait to be called upon.#' 
[21:02] <@breeden> Ok, so before we start, I think it would be a 
good idea to discuss what hochs, i_c-Y, and I are trying to do here.
[21:03] <@breeden> We are ultimately trying to make efnet.#math 
more of an interactive environment that encourages learning and sharing 
with others.
[21:04] <@breeden> We would ultimately like for this reading 
seminar to be fairly informal, and not to follow that standard lecture 
feel.
[21:05] <@breeden> but since this is the first week, and everyone 
really doesn't know what is going on, hoch and I are just going to give a
 lecture on topics that can be found in the first chapter of Bredon's 
Topology & Geometry
[21:06] <@breeden> Afterwards, if everyone hasn't fallen asleep 
already, we will try to get some input from the audience on how we 
should proceed.
[21:07] <@breeden> ok, so on the note I guess we will 
start :)
[21:08] <@hochs> so the topic you will discuss is nets and 
tychonoff for the week 1 talk 
[21:08] <@breeden> Today I will be talking about an idea that is 
found in General Topology that tries generalize sequential arguments 
found that are often found when working with metric spaces.
[21:09] <@breeden> When we are working with metric spaces, we able
 to characterize many important topics using sequences such as 
continuous functions, compactness, and the closure of a subspace.
[21:09] <@breeden> However, outside of the world of metric spaces,
 sequences are not longer powerful enough to make the same 
characterizations.
[21:10] <@breeden> Essentially, it was found that sequences didn't
 have the cardinality to really exploit an arbitrary topological space.
[21:11] <@breeden> Nets were later introduced to fix this.  With 
nets we will be able to make the same characterizations we did in metric
 spaces for general topological spaces.
[21:11] <@breeden> Characterize continuous functions, compactness,
 and many other things
[21:12] <@breeden> Also, we will be able to give a very slick 
proof of Tychonoff's theorem using nets.
[21:12] <@breeden> So, let us recall that in a topological space X
 a map f: N -> X, where N denotes the natural numbers, is called a 
sequence.
[21:13] <@breeden> For notational convenience we use x_n instead 
of f(n)
[21:13] <@breeden> The generalization of this, will allow for 
higher cardinality, and even remove the restriction that N is ordered.
[21:14] <@breeden> We will replace N by what is called a direct 
set.
[21:14] <@breeden> A directed set, D, is a partially ordered set 
such that for any two elements x and y in D, there is an element z with z
 >= x and z >= y.
[21:15] <@breeden> We don't require D to be ordered, but only 
partially ordered.  And as for as far as I can see, this just will make 
our constructions of directed sets easier.
[21:15] <@breeden> Now, we will call a net a map f: D -> X, 
where D is a directed set and X is a topological space
[21:15] <@breeden> Oh.. and btw, we removed the ! for 
questions thing, so if anyone has any questions just ask :)
[21:16] <@breeden> Ok, to facilitate the notation a little bit, we
 will need a few more definitions.
[21:16] <@breeden> However, don't fret, these are equivalent to 
the definitions you see concerning sequences.
[21:17] <@breeden> first, we will, for the rest of the seminar, 
use the notation x_a to me f(a)
[21:17] <@breeden> and hopefully there will be no confusion as to 
what f is :)
[21:18] <@breeden> Now, for x_n is a net, and A is a subset of X, 
we say that x_n is "frequently" in A if for any a in D there is a b 
>= a such that x_b \in A.
[21:19] <@breeden> And we say that a x_n is eventually in A if 
there exist an a in D such that for every b >= a then x_n is in A
[21:19] <@breeden> And finally, we say that x_n converges to x if 
for any open set U containing x, then x_n is eventually in U.
[21:20] <@breeden> Now, let's try to see some nets in action.
[21:20] <@breeden> This theorem will give us an idea of how nets 
allow use to essentially "distinguish" points in a toplogical space.
[21:21] <@breeden> Recall, we say a topological space X is 
hausdorff if for any x and y in X we can find open sets U and V 
containing x and y respectively where U n V is empty.
[21:21] <@breeden> Theorem.  A topological space X is hausdorff if
 and only if any convergent net has a unique limit.
[21:22] <@breeden> Now, most of as familiar that being hausdorff 
implies uniqueness of limits.
[21:22] <@breeden> And it's not necessarily true that if all 
sequences in X have a unique limite then X is hausdorff
[21:23] <@breeden> But this shows that going to nets is the right 
generalization to detect being hausdorff
[21:23] <@breeden> So let's see how this proof goes.
[21:24] <@breeden> We will show this using contradiction (or 
contraposition), by showing that if X is not Hausdorff then we can find a
 net that doesn't have unique limits.
[21:24] <@breeden> So, suppose that X is not Hausdorff.  So there 
exists x and y in X that can't be separated.
[21:25] <DaMancha> can i interrupt for a second?
[21:26] <@breeden> This means that for any open sets x in U and y 
in V, then we have U n V is non empty.
[21:26] <@breeden> So we want to construct a net that has two 
distinct limit points.  First we need to construct a directed set.
[21:26] <@breeden> Let D denote the collection of pairs (U,V) of 
open sets such that x in U and y in V.
[21:26] <@breeden> We can impose a partial ordering on D by (W,Z) 
>= (U,V) if W subset U and Z subset V
[21:26] <@breeden> oh phone
[21:26] <@breeden> yup
[21:27] <DaMancha> e the collection of pairs (U,V) of open sets 
such that x
[21:27] <DaMancha>           in U and y 
[21:27] <DaMancha> crap
[21:27] <DaMancha> you mentioned the relative inadequacy of 
sequences (which motivates nets).
[21:27] <@breeden> yes
[21:27] <@breeden> would you like to see an example?
[21:27] <DaMancha> this was a very important concept.
[21:27] <DaMancha> one aspect is they're short since they're 
indexed on N. however, even if we were to index them on something bigger
 than N they'd still come up short, right?
[21:28] <@breeden> short for this theorem?
[21:28] <@breeden> Well in fact, the cardinality for the index 
required is intimately related to the cardinatliy of a basis at a point
[21:29] <DaMancha> in general, to characterize fully the toplogy
[21:29] <@breeden> for instance, if we were to insist that each 
point in X had a countable basis, then the theorem im proving would hold
 true for sequences in place for nets
[21:29] <DaMancha> sorry, go ahead, i just thought it was worth 
hammering why sequences were inadequate and why we were happy with the 
advent of nets.
[21:30] <@hochs> cardinality isn't the only issue probably.  you 
also need to lose "total" in "total order" and get more tree-ish 
structures to make the proof go through 
[21:30] <@breeden> that is true, at least for some of the later 
theorems
[21:30] <@breeden> but for these two theorems i don't think that 
is needed
[21:31] <Polytope> so transfinite sequences would also be 
inadequate?
[21:31] <@breeden> i think it would fail to characterize 
compactness, at least in the context that is presented in Bredon's
[21:31] <@hochs> does anyone know a quick/easy example of a 
non-hausdorff space where every sequence converges uniquely? (if it 
converges)
[21:31] <@breeden> yes
[21:31] <@breeden> Take the co-contable topology on R
[21:32] <@breeden> that is U is open in R if and only if the 
complement of U is countable.
[21:32] <@breeden> Now take a sequence in x_n converging to x.
[21:32] <@breeden> Then x_n must eventually be in (R - { x_n }) u {
 x}
[21:33] <@breeden> so the sequence must eventually be constant
[21:33] <@hochs> i see, thank you 
[21:34] <@breeden> but yeah, that topology is no hausdorff to the 
least bit
[21:34] <@breeden> ok, let's finish this proof really quick
[21:35] <@breeden> it's our first sighting of a net, and may help 
us give intuition of how they work
[21:35] <@breeden> Ok so we had a non-hausdorff space and points x
 and y that couldn't be separated
[21:36] <@breeden> then we defined a directed set D of collection 
of pairs (U,V) of open sets containing x and y respectively.
[21:36] <@breeden> And we imposed the partial ordering (W,Z) >=
 (U,V) if W subset U and Z subset V
[21:36] <@breeden> And we noted that U n V is never empty since x 
and y couldn't be separated.
[21:37] <@breeden> ok, now we define our net
[21:37] <@breeden> We define our net so that x_{(U,V)} is in U n V
[21:38] <@breeden> and we claim that x_{(U,V)} converges to both x
 and y.  Ok, so let U be an open set contain x.  Then for any (W,Z) 
>= (U,X) we have that x_{(W,Z)} \in W n Z \subset U n X = U.
[21:38] <@breeden> so x_{(U,V)} is eventually in U and hence 
converges to x, and similarly for y.
[21:39] <@breeden> and that's the proof
[21:39] <@breeden> Right, so I wanted to make a point here.  That 
if we did assume that each point of x had a countable basis, then we 
could replace arbitary open subsets of x and y to basis elements.
[21:40] <@breeden> Now if we had U_n as a decreasing sequence of a
 basis for x and V_n as a decreasing sequence for a basis of V_n
[21:40] <@breeden> we could define f(n) \in V_n n U_n
[21:41] <@breeden> to construct a sequence that converges to both x
 and y, and achieve the same result.
[21:42] <@breeden> And this same observation will also hold true 
for the characterization of continuous functions.
[21:42] <@breeden> we can use sequences, like we did with metric 
spaces, if we assume each point has a countable basis.
[21:42] <@breeden> Let's see how.
[21:43] <AreEssay> Just a quick question: what did you mean by 
nets allow us to distinguish points in a space?
[21:43] <@breeden> at the very beginning?
[21:43] <AreEssay> yeah
[21:43] <DaMancha> can you please re-state the theorem too?
[21:44] <@breeden> sure
[21:44] <@breeden> let me restate the theorem first
[21:44] <@breeden> Theorem.  X is hausdorff <=> every net 
has a unique limit.
[21:44] <@breeden> I neglected on direction, and only showed the 
part where we get to see nets in action
[21:45] <DaMancha> i see. so nets are useful since sequences can 
have unique limits in non-hausdorff spaces so their limiting behavior 
doesn't fully characterize hausdorfness (sorry for the makeshift lingo)
[21:45] <@breeden> AreEssay, this is what I meant essentially.  
Being able to detect a space being Hausdorff, i guess.
[21:45] <@breeden> DaMancha, yup, that's how i see it too
[21:46] <AreEssay> oh, neat
[21:46] <@hochs> a quick google search returned 
http://math.stackexchange.com/questions/100380/is-a-net-stronger-than-a-
transfinite-sequence-for-characterizing-topology  (polytope's unanswered
 question) - just a side remark for anyone who is interested in this 
sort of thing
[21:46] <@breeden> what did they say?
[21:46] <@hochs> from a quick glance, it seems the op asked if 
transfinite sequences is enough 
[21:46] <Polytope> it is stronger. something about stone-cech 
compactifications
[21:46] <@hochs> yea
[21:47] <@breeden> nets are stronger?
[21:47] <@breeden> oh right
[21:47] <@breeden> in the example i gave, i think if we indexed by
 R, we would be ok!
[21:48] <@breeden> i guess i'll continue
[21:48] <@breeden> So, we can show that nets can characterize 
continuous functions.
[21:49] <@breeden> theorem.  A map f: X -> Y is continuous 
<=> f(x_n) converges for every convergent net x_n.
[21:49] <@breeden> something that we all learned in calculus one.
[21:49] <@breeden> for the (=>) direction it goes something 
like this:
[21:50] <@breeden> oh wait
[21:50] <@hochs> do you need f(x_n) converging to f(x), for x_n 
converging to x?
[21:50] <@breeden> i think i need that f(x_n) converges to f(x) 
there
[21:50] <@breeden> yeah
[21:50] <@breeden> :)
[21:50] <@breeden> i dont know if those two statements are 
equivalent or not
[21:51] <@breeden> theorem.  A map f: X -> Y is continuous 
<=> f(x_n) converges to f(x) for every net x_n converging to x.
[21:51] <@breeden> Right, so suppose that f is continuous, and let
 x_n -> x.  Let U be an open set containing f(x).  then f^{-1}(U) is 
open, and x_n is eventually in f^{-1}(U) since x_n -> x \in 
f^{-1}(U).
[21:52] <@breeden> But then f(x_n) is eventually in U.
[21:52] <@breeden> So f(x_n) converges to f(x).
[21:52] <DaMancha> sorry, i missed a second or two.
[21:52] <@breeden> what was the last thing you saw?
[21:53] <DaMancha> we have that f is cont. iff for all nets that 
converge to x we have f(net) converge to f(x) ?
[21:53] <@breeden> yes
[21:53] <DaMancha> ok, thanks.
[21:53] <@breeden> yup :)
[21:53] <@breeden> ok, for the other direction we will use 
contradiction.
[21:53] <@breeden> (well contraposition)
[21:54] <@breeden> Suppose that f is not continuous.  We will 
construct a new x_n that converges to x, but f(x_n) doesn't converge to 
f(x).
[21:54] <@breeden> Since f is not continuous, there exist an open 
set U such that f^{-1}(U) is not open.
[21:55] <@breeden> that means we can find a boundary point on 
f^{-1}(U), let's call this x
[21:56] <@breeden> Now, let D be all open sets containing x, and 
order it by A >= B if A subset B.
[21:56] <@breeden> and define x_A to be in A for each A in D.
[21:56] <@breeden> eh, i mean, to be in A - f^{-1}(U) :)
[21:57] <@breeden> this is always possible because x is a boundary
 point so A - f^{-1}(U) is non-empty.
[21:57] <DaMancha> ah but you get convergence into f^-1(U)?
[21:57] <@breeden> Now, by construction we see that x_A converges 
to x, but f(x_A) is not in U for any A
[21:58] <@breeden> and that's the proof
[21:58] <@breeden> Again if we had a countable basis, we could 
replace the nets with sequences :)
[21:59] <@breeden> ok, so the rest is more involved.
[21:59] <@breeden> do you guys think we should carry on talking 
about nets?
[22:00] <@breeden> essentially, we our next goal would be to show 
that, X is compact <=> Every net has a convergent subnet.
[22:00] <@breeden> and from this we can quickly prove Tychonoff
[22:00] <@hochs> bredon likes to use universal subnet as an 
intermediary for that theorem 
[22:01] <@hochs> though you don't really need it
[22:01] <@breeden> is this not standard?
[22:01] <@hochs> not that i know of - maybe it is standard, i 
don't know
[22:01] <@hochs> but folland, for example, has a proof where he 
just does it without universal subnet
[22:01] <@breeden> using nets?
[22:01] <@hochs> yes
[22:01] <@breeden> huh, interesting
[22:02] <@breeden> let's just introduce universal nets and show 
that every net has a universal subnet without proof to cut time
[22:02] <@breeden> ok, so first you need to have an understanding 
of what subnet is
[22:03] <@breeden> for sequences, if f: N -> X is a sequence, 
and g: N -> N is increasing, then f(g(n)) is a subsequence
[22:03] <@breeden> nets use something similar
[22:04] <@breeden> instead of having an increasing map, we use 
something called a final map.
[22:04] <@breeden> That is, given a directed set D, a map is a map
 h: D' -> D, such that for any d in D we can find a d' in D' such 
that a >= d' implies that x_{h(a)} >= d
[22:05] <@hochs> so increasing N -> N is a final map 
[22:05] <@breeden> eh
[22:05] <@breeden> h(a) not x_{h(a)}
[22:05] <DaMancha> ok, so the order here is applied to the 
function from D to ...
[22:05] <@hochs> isn't it x_{h(a)}?
[22:05] <AreEssay> D' is directed too?
[22:05] <DaMancha> vs the sequence where you have monotonicity
[22:05] <@breeden> oh right, D' is directed to
[22:05] <@hochs> err sorry, h(a)
[22:06] <@breeden> right so subsequences are subnets by this 
definition
[22:07] <@breeden> so, to see this in action we have
[22:07] <@breeden> a net x_n has a subnet that converges to x in X
 <=> for any open set U, x_n is frequently in U.
[22:07] <DaMancha> right N is a directed set. ok, so a 
generalization on what we know as subsequences
[22:08] <@hochs> <@breeden> Now, for x_n is a net, and A is a
 subset of X, we say that x_n is "frequently" in A if for any a in D 
there is a b >= a such that x_b \in A.
[22:08] <@hochs> to remind what frequently means
[22:09] <@breeden> Ok, we will show the <= direction only, 
because that's the only interesting part
[22:09] <@breeden> So let x_n be a net such that for any open set U
 containing x, then x_n is frequently in U.
[22:09] <@breeden> Now, there are two things we will need to keep 
track of when constructing our net.
[22:10] <@breeden> our subnet*
[22:10] <@breeden> we will need to make sure that it is eventually
 in any open set neighborhood of X, and we will need to keep in mind 
what our final map will be.
[22:11] <@breeden> So let D' be the collection of pairs (n, U) 
where n is in D and U is a neighborhood of x
[22:11] <@breeden> and such that x_n is in U.
[22:11] <@breeden> Now, we can define the map h: D' -> D (where
 D is the index fo x_n), by h(n,U) = n.
[22:12] <@breeden> We can order D' by (b,V) >= (a,U) if b >=
 a and V subset V.
[22:12] <@breeden> now, this defines a final map.
[22:13] <@breeden> And further more, for any U containing x, we 
can find (n, U) in D', where x_{h(n)} is in U
[22:13] <@breeden> and for any (s, V) >= (n, U) you have 
x_{h(n)} in V \subset U, which shows convergence.
[22:14] <@breeden> so that shows that if x_n is frequently in U, 
then x_n has a subnet that converges to x.
[22:14] <@breeden> ok, we are almost there.
[22:15] <@breeden> We still want to characterize compactness for 
topological spaces using nets
[22:15] <@breeden> But first, we need one idea
[22:15] <@breeden> I like to think of this as an analogue to 
Cauchy sequences for metric spaces, but that is not entirely accurate.
[22:16] <@breeden> However, it will almost take the place of 
Cauchy sequences in our characterization of compactness
[22:16] <AreEssay> Quick question:
[22:16] <@breeden> yes?
[22:16] <AreEssay> Can you perhaps comment on the difference 
between a net being frequently in a subspace A of X, rather than being 
eventually in A. Is the notion of being eventually in a subspace 
stronger?
[22:16] <@breeden> yes
[22:16] <@breeden> much stronger
[22:16] <AreEssay> ok
[22:17] <@breeden> in the context of sequences, x_n is frequently 
in A if for any M we can find n >= M such that x_n is in A.
[22:17] <@breeden> think of x_n = (-1)^n
[22:17] <@breeden> this sequence is frequently in { 1 } and 
frequently in { -1 }
[22:17] <@breeden> but it never stays in either of them.
[22:18] <@breeden> Eventually means that there is an M such that 
x_n is always in A for n >= M.
[22:18] <@hochs> equivalent of "has a *subsequence* which 
converges to x" 
[22:18] <AreEssay> great, thanks!
[22:18] <@breeden> so x_n = 1 is eventually in { 1 }, because it 
never leaves.
[22:19] <@breeden> ok
[22:19] <@breeden> so we say that a net x_n is "universal" if for 
any A \subset X then x_n is eventually in A or eventually in X - A.
[22:19] <@breeden> It's worth noting that for any A \subset X, you
 will have one of three things
[22:19] <@breeden> (i) x_n is eventually in A, or x_n is 
eventually in X - A
[22:20] <@breeden> or (ii) x_n is frequently in both A and X - A
[22:20] <@breeden> a universal net is one where (ii) never 
happens.
[22:20] <@breeden> Now, there is a theorem that says every net has
 a universal subnet.
[22:21] <@breeden> but i don't think i'll go over the proof, 
because i'm already 20 minutes late :)
[22:21] <@breeden> and it's quite contrived
[22:21] <DaMancha> what's the third thing?
[22:21] <@breeden> yeah, (i) counts as two
[22:22] <@breeden> but anyways, let's just take this as a 
technicality and prove the characterizatoin of compactness
[22:22] <@breeden> Recall, a space X is compact if every open 
covering of X has a finite subcovering.
[22:23] <@breeden> there is an equivalent definition that uses 
closed sets instead of open sets, and this will be useful here.
[22:23] <@breeden> The equivalent definition is: X is compact 
<=> Every collection of closed sets having the finite intersection
 property has non-empty intersection.
[22:24] <@breeden> The finite intersection property of a 
collection of closed sets is: Any finite intersection is non-empty.
[22:24] <@breeden> so essentially it's saying that, if any finite 
intersection is non-empty, then the entire collection has non-empty 
intersection.
[22:25] <@breeden> and the theorem:
[22:25] <@breeden> Theorem: X is compact <=> Every universal
 net in X converges <=> Every net in X has a convergent subnet.
[22:26] <@breeden> ok, so let's see how this goes.
[22:26] <@breeden> (i) => (ii)
[22:26] <@breeden> Suppose that x_n is a universal net.  For the 
sake of a contradiction, suppose that x_n does not converge.
[22:26] <@breeden> Then for any x in X we can find a neighborhood 
U_x of x, such that x_n is eventually in X - U_x
[22:27] <@breeden> this is because x_n is universal, and we must 
have that x_n is eventually in U_x or X - U_x
[22:28] <@breeden> ok, that means there exists a b_n such that a 
>= b_n => x_a is not in U_x.
[22:28] <@breeden> By the compactness of X we can find a finite 
subcover, X = U_{x_1} u U_{x_2} u ... u U_{x_n}.
[22:29] <@breeden> i made bad notation here.
[22:29] <@breeden> b_n was suppose to be b_x up there.
[22:29] <@breeden> ok, that means there exists a b_n such that a 
>= b_x => x_a is not in U_x.
[22:29] <@breeden> there exists a b_x such that...
[22:29] <@breeden> ok, the point is we can find b >= b_{x_i} 
for each i
[22:30] <@breeden> (like taking the maximum).
[22:30] <@breeden> But then we would have that x_b is not in 
U_{x_i} for any i, which means that x_b is not in X.
[22:30] <@breeden> This is a contradiction.
[22:31] <@breeden> (ii) => (iii) is pretty easy.  It follows 
from that theorem that ever net has a universal subnet, and by 
assumption every universal net converges.
[22:31] <@breeden> (iii) => (i)
[22:31] <@breeden> We will how that X is compact using the 
closed-set definition of compactness.
[22:32] <@breeden> Now let F = { C } be a collection of closed 
sets satisfying the finite intersection property.
[22:32] <@breeden> We can assume that F is closed under 
intersection, since this won't effect the intersection of F.
[22:32] <@breeden> closed under finite intersection* sorry :)
[22:33] <@breeden> Then we can make F into a directed set by using
 the (standard) ordering, C' >= C if C' \subset C.
[22:33] <@breeden> Now we need to find an element that is in each 
closed set C in F.
[22:34] <@breeden> For each C in F, we define x_C \in C (any 
element will do).
[22:34] <@breeden> But assumption, we know there is a convergent 
subnet.
[22:34] <@breeden> That is, there is a final map h: D -> F, for
 some directed set D.
[22:34] <@breeden> Now, suppose that x_{h(n)} -> x, and let C 
in F.
[22:35] <@breeden> By the definition of being a subnet, there 
exists b in D such that a >= b implies h(a) >= C implies h(a) 
subset C.
[22:35] <@breeden> (recall that h(a) is a set)
[22:35] <@breeden> since sets are our index, which is often the 
case for nets.
[22:36] <@breeden> Now this says that x_{h(n)} \in h(n) \subset C
[22:36] <@breeden> and so x_{h(n)} is eventually in C.
[22:36] <@breeden> Since C is closed, it contains all of it's 
(net) limit points (by something i neglected to mention earlier, heh)
[22:37] <@breeden> and so x (the limit point of x_{h(n)} is in C).
[22:37] <@breeden> and since C was arbitrary, you see that x is in
 every C is F.
[22:37] <@breeden> which shows that X is compact.
[22:37] <@breeden> did everyone fall asleep?
[22:38] <@hochs> nope
[22:38] <AreEssay> no
[22:38] <@breeden> :)
[22:38] <@breeden> ok
[22:38] <@breeden> so that is a really cool characterization of 
compactness
[22:38] <@breeden> and let's show an easy application.
[22:39] <@hochs> which has a characterization in terms of 
sequences for metric spaces right 
[22:39] <@breeden> right, for metric spaces it is: Every sequence 
has a convergent subsequence.
[22:39] <@hochs> so this again sort of captures the idea of 
generalizing this characterization from second countable normal spaces 
to general spaces
[22:39] <@hochs> (second countable normal spaces are 
metrizable :) )
[22:39] <@breeden> hehe
[22:40] <@breeden> idk if this would still hold for first 
countable spaces
[22:40] <@breeden> i dont think so, gut feeling
[22:41] <@breeden> ok, and will all this theory, we can give a 
really slick proof of Tychonoff's Theorem.
[22:41] <@breeden> Tychonoff's theorem.  An arbitrary product of 
compact spaces are compact.
[22:41] <@breeden> first, we need to know a little something about
 product spaces.
[22:42] <@breeden> Given topological spaces V_n, we define their 
product X V_n, to have the strongest topology such that their 
projections are continuous.
[22:42] <@breeden> what do i mean by that?
[22:42] <@breeden> a projection is a map f_k: X V_n -> V_k
[22:43] <@breeden> that takes the element in the k position and 
sends it to it's rightful owner in V_k
[22:43] <@breeden> for f_k to be continuous we need that for any 
open set U in V_k that f_k^{-1}(U) is open in X V_n
[22:44] <@hochs> and take finite intersections of those?
[22:44] <@hochs> to form a basis
[22:44] <@breeden> we an take this a sub-basic, meaning that we 
have a basis of all finite intersections of those guys
[22:44] <@breeden> yeah
[22:45] <@breeden> which means all open sets are unions of all 
finite intersections of those guys ;)
[22:45] <@breeden> ok, but what's the point?
[22:45] <@breeden> the point is this property:
[22:46] <@hochs> Q: what kind of topology is the really big one 
that has as its basis arbitrary (not just finite) intersections? 
[22:46] <@hochs> is it compact?
[22:46] <@breeden> that would be discrete no?
[22:46] <@hochs> not necessarily, but that wouldn't be compact 
[22:46] <@breeden> as for nice X
[22:47] <@breeden> ie, if x is in intersection of all opensets 
that contain x
[22:47] <@breeden> { x } is the ...
[22:48] <@hochs> that sounds suspiciously close to total 
disconnectedness 
[22:48] <@breeden> I mean for an arbitrary product of R, then the 
topology you mentioned would give the discrete topology?
[22:48] <@hochs> at least for compact spaces, the connected 
component of {x} is the intersection of all the open & closed nbhds 
of x 
[22:48] <@hochs> or locally compact
[22:49] <@hochs> yes, you are right 
[22:49] <@hochs> so they're not very interesting :) 
[22:49] <@breeden> hehe
[22:49] <@breeden> ok, so the point is x_n converges to x in X V_n
 if and only if f_k(x_n) converges to f_k(x) for all k.
[22:50] <@breeden> that is, it converges if and only if it 
converges in each component
[22:50] <@breeden> and it's not hard to show this
[22:50] <@breeden> does anyone want to see a proof?
[22:50] <@breeden> great, so here is one:
[22:50] <@breeden> (we'll just show one direction)
[22:51] <@breeden> suppose that f_k(x_n) -> f_k(x) for all k.
[22:51] <@breeden> let U be a basis element for x in X V_k
[22:51] <@breeden> then U = X B_k where B_k = V_k for all but 
finitely many indicies
[22:52] <@breeden> say that k_i are the indicies where you dont 
have B_{k_i} = V_{k_i}
[22:53] <@breeden> then for each k_i there is a n_i such that a 
>= x_{n_i} implies that x_a is in U.
[22:53] <@breeden> take M >= n_i for each i,
[22:54] <@breeden> then for each a >= M you have that x_a is in
 U, and for all the other indicies, it's trivially because B_k = V_k
[22:54] <@breeden> which shows convergence.
[22:54] <@breeden> bah, luckly matt isn't here :)
[22:54] <@cyby> ?
[22:54] <@breeden> i said trivially
[22:54] <@hochs> :)
[22:54] <@cyby> (I'm sorry, I hope someone is logging this, btw, 
so it can be posted)
[22:55] <@breeden> he gives me shit for adding ly :)
[22:55] <@hochs> i'm logging this 
[22:55] <@cyby> Great.
[22:55] <@breeden> ok, great
[22:55] <@breeden> so x_n converges to x in X V_n if and only if 
each component dos
[22:55] <@breeden> does*
[22:55] <@breeden> so now we're ready for Tychonoff's theorem.
[22:55] <@breeden> That is, X V_k is compact if each V_k is 
compact.
[22:56] <@breeden> By our characterization of compactness, we just
 need to show that every net has a convergent subnet.
[22:56] <@breeden> whoops
[22:56] <@breeden> im missing something here, do you remember the 
argument hochs?
[22:57] <@hochs> i think bredon used universal nets 
[22:57] <@breeden> i think so too
[22:57] <@hochs> every universal net converges <=> X is 
compact 
[22:57] <@breeden> let me get my copy of bredon
[22:58] <@hochs> so you take any universal net x_n in X, then the 
projections are also all universal nets
[22:58] <@breeden> heh, this was supposed to be the easiest part, i
 must be getting tired :)
[22:58] <@hochs> which converge (by the characterization of 
compactness by nets again)
[22:58] <@breeden> oh right, that's it
[22:58] <@breeden> qed :)
[22:58] <@hochs> that's a very short proof, just like the proof of
 fundamental theorem of algebra using liouville's theorem - complex 
analysis
[22:59] <@hochs> but you have plowed hard with nets 
[23:00] <@breeden> all of the work went into a theorem we didn't 
prove
[23:00] <@breeden> the existence of universal subnets
[23:00] <@hochs> as you said, the universal subnet thing has a 
contrived proof
[23:00] <@breeden> you can convince yourself of what it's the 
right thing to do
[23:00] <@breeden> something along the lines:
[23:01] <@breeden> For any set A in X either x_n will eventually 
be in A or X - A, or x_n will be frequently in both
[23:01] <@breeden> and this characterization, you can try to find a
 subnet by using magic
[23:02] <@hochs> right, so the indexing sets we have used all look
 like either 1) certain open subsets of the space, or 2) pairs (n,U) of 
original directed set and U certain open subset
[23:02] <@hochs> all we really needed i think 
[23:02] <@hochs> is that right?
[23:03] <@breeden> right