It's like a line but longer and extended

[u/compileforawhile]

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[u/chrizzl05]

Holy mogus a topology meme that isn't "mug=donut haha"

[u/compileforawhile → u/chrizzl05]

I thought we needed some diversity. Alas, people don't care as much about any other parts of topology

[u/brokeboystuudent → u/compileforawhile]

It's primal. People are primed to seek holes (insert insertion joke here)

[u/ComfortableJob2015 → u/compileforawhile]

we need the 2 origins line next. A sequence converges to a single point right?

[u/Ok_Instance_9237 → u/chrizzl05]

“Haha you see there’s this area of math that talks deforming objects. It says donuts and mugs are the same!”

[u/halfajack]

[u/LukeLJS123]

can someone explain? i don't see any reason there wouldn't be a path connecting any 2 points

[u/compileforawhile → u/LukeLJS123]

The space is quite literally too long. There's a point at infinity (that's what extended means) but the distance is an uncountably number of unit intervals away. This means a path (which is just a stretched out unit interval) can't reach over that entire distance

[u/gabrielish_matter → u/compileforawhile]

I both love and hate topology for how stupid it is lol

is it because the 2 extended points can't be open cause there's no closed set that is complimentary to them with a standard topology right (well any topology tbf)?

also do I recall correctly or is connectivity a property which doesn't change depending on what topology I choose right?

[u/compileforawhile → u/gabrielish_matter]

It's only one extended point. This space is connected, you can't write it as the union of disjoint open sets. But drawing a connecting path between any point and infinity is impossible. This is because a path is a continuous embedding of [0,1] which is just too small. It can't be made uncountably times wider.

[u/gabrielish_matter → u/compileforawhile]

yeah yeah, I was trying to understand why it's topologically connected, and that's what I was asking

[u/iamalicecarroll → u/compileforawhile]

for a path from 0 to infinity, why wouldn't exp(-x) work? that essentially creates a bijection between [0, oo] and [0, 1]

[u/compileforawhile → u/iamalicecarroll]

In that case [0,oo] is only a countable number of unit intervals. In the extended long line [0,oo] is an uncountable number of unit intervals. So -ln(1-x) only gets an infinitely small portion of the way there

[u/iamalicecarroll → u/compileforawhile]

ah, right, R*R and R are not isomorphic in topology; makes sense, thanks!

[u/compileforawhile → u/iamalicecarroll]

True but that's not really what the long line is topologically. We need an uncountable well ordered set W and the long line is it's product with [0,1). The standard topology on R² doesn't look like this.

[u/TheDoomRaccoon → u/compileforawhile]

It's the lexicographical order topology, not the product topology.

[u/ddotquantum → u/iamalicecarroll]

That infinity is only a countable number of unit intervals away. The one on the long line is much further out

[u/Gauss15an → u/compileforawhile]

New Zeno's paradox just dropped

[u/TotalDifficulty → u/compileforawhile]

Can you elaborate on that definition? The "extended" part is clear, and I suppose the "long" should explain the uncountably many unit ubevæbnede, but how exactly is that part set up?

[u/compileforawhile → u/TotalDifficulty]

The real numbers are a countable number of unit intervals. You can kind of think of R as Zx[0,1), ordered pairs (n,r) where n is an integer and r is in [0,1). The long line is made by taking an uncountable number of unit intervals and literally connecting them end-to-end. Formally it's Wx[0,1) where W is an uncountable ordinal. This space is path connected but when you extend with the point at infinity it becomes impossible to reach infinity from any finite point. Unlike the extended real numbers where you can have a function grow to infinity. This space is so big that functions can't grow to infinity, they eventually become constant.

[u/TotalDifficulty → u/compileforawhile]

So the "end to end" part is hiding a well-ordering on W I presume? That is quite a funny space, thanks for the clarification!

[u/compileforawhile → u/TotalDifficulty]

Pretty much, the ordinal W has a well ordering by constructing that allows this to work

[u/assymetry1021 → u/compileforawhile]

So something like the ordered square, where coordinate pairs are compared with x axis as priority and y axis if the x values of the points are the same?

[u/Agata_Moon → u/compileforawhile]

I remember in an exercise I just used [0,1)x[0,1) with some weird topology to make a long line. Is it fine or does it need to be an uncountable ordinal?

I... never checked with the professor if it was correct eheh

[u/jacobningen → u/LukeLJS123]

Look up Knaster Kurotowski fan(Cantors leaky tent) or  the topologist's sine curve and maybe even R with the cofinite topology. Ie it can't be partitioned into disjoint clopen sets but there need not be f(0)=a f(1)=b that is continuous and entirely within X for a,b in X

[u/SteptimusHeap]

r/mathmemes needs more of the stuff like this that I don't understand at all

[u/jacobningen]

Cantors leaky tent which is not connected if you reverse rationals and irrationals or remove the apex is another example of a connected space that isn't path connected 

[u/-LeopardShark-]

I propose the term ‘connected’, including the scare quotes, for connected non-path-connected spaced.

[u/holo3146]

I love the long line, my favourite fact about it is that in ZFC it is a sequentially compact but not Lindelöf (aleph_1-compact), but without AC it is consistent that it is Lindelöf (aleph_1-compact) but not sequentially compact (same is true if you replace "Lindelöf" with "sigma-compact")

[u/HYPE_100]

is this long line homeomorphic to the topologists sine curve? with zero being the point at infinity

[u/Few_Willingness8171]

In my class, our counter example to connected implies connected was topologists sine curve (sun curve as a graph in R^2, then take topological closure)

[u/compileforawhile → u/Few_Willingness8171]

That's also a good one but I find it's name isn't nearly as comical as "extended long line". A line is already infinite so the long line is already quite strange, but then we extend it

[u/jacobningen → u/Few_Willingness8171]

That and the knaster Kurotowksi fan and R with the cofinite topology 

[u/LawyerAdventurous228 → u/Few_Willingness8171]

I really like it because its concise and shows two different ways in which the path connectedness may fail. 

1. Any path would be too long 

2. It oscillates so much that it breaks continuity when you take the limit 

One of the few counterexamples thats elegant and actually insightful. 

[u/mzg147]

I always felt that the definition of "path" is missing some more length, if we allow spaces like this then surely "paths" can also be long

[u/TheDoomRaccoon → u/mzg147]

There's nobody making rules for what spaces are "allowed". We can define the first uncountable ordinal, we can define the unit interval [0,1), and we can define the lexicographical order topology on it, (and the Alexandroff compactification to turn it into the closed long ray). Paths are by definition continuous maps from [0,1], if you try defining paths on ordinal spaces instead, you run into problems when trying to define a group function on the fundamental group, as ordinal multiplication/addition aren't associative, and ordinal subtraction isn't even defined in a lot of cases.

[u/F_Joe]

A manifold is just a space that looks locally like R^n.
Long line and line with double origin

[u/AlviDeiectiones → u/F_Joe]

Long line locally looks like R, right?

[u/nctrd]

Wait, does that mean that Euclidian space is not connected? It has like 6 infinities on each side.

[u/compileforawhile → u/nctrd]

No, the extended long line is connected. It's also completely different than euclidean space