
Long line (topology) - luu
https://en.wikipedia.org/wiki/Long_line_(topology)
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marvel_boy
Newbie here. What is the counterexample of Topolopy Long Line serves?

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fantispug
It is a Hausdorff space that is locally homeomorphic to the Euclidean plane,
but not paracompact. Manifolds are usually defined to be paracompact because
this implies the existence of partitions of unity which have many nice
implications: [http://math.stackexchange.com/questions/98105/why-are-
smooth...](http://math.stackexchange.com/questions/98105/why-are-smooth-
manifolds-defined-to-be-paracompact)

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jordigh
Euclidean plane? Not line? Did I misunderstand the construction?

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contravariant
No, he probably meant the euclidean line.

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kazinator
Insert "come from a long line of topologists" jokes here.

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airza
it's really weird to see a euclidian-looking line that has no strictly
increasing sequences or functions..

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olivermarks
I am baffled by this and feeling stupid

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gjulianm
I wouldn't worry about that. I'm studying mathematics and I also feel stupid
while reading new concepts on Wikipedia. The problem with mathematics is that,
if you don't know the idea behind the concept or which kind of problem is it
trying to solve, it's almost impossible to really understand the definitions.

And even when you understand the concepts and ideas, it can still be difficult
to understand the definitions. Few months ago, a professor told us the 100%
formal definition of a function: it wasn't easy to understand and we all had
this "wtf" face.

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jordigh
Yeah, the thing with formal definitions is that they're like little programs
you have to compile before you can see their output. Trying to understand what
a function is from a statement such as "a function from A to B is a subset F
of the cartesian product AxB such that for all (a1,b1), and (a2,b2) in F,
b1=b2 whenever a1=a1" really obscures the notion this is all about "mapping"
or "doing" something. This definition is more like something that you would
feed to a computer who doesn't "understand" what really is going on. Or to a
human, who would work through it piecemeal, and try to build a few functions
following this definition until the a-ha moment would come and this human
would form a different intuitive understanding of what a function is or does.

But the reason we write definitions as little programs to be compiled is to
make sure that there can be no room for ambiguity. You compile the definition
and arrive at the same mathematical object as anyone else. If we define a
function as a mapping or a procedure, then we just shifted one word for
another and still have to elucidate what a mapping or a procedure is.

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ihm
Well, the set theoretic attempt to describe functions isn't a very nice one.

Type theory gives a formal definition closer to the intuitive notion: a
function from A to B is a program of type A -> B. I.e., a rule for turning
things of type A into things of type B.

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jordigh
I admit I don't know much about type theory. I assume "program" is a term of
art? Also, does this allow for functions that have literally undescribable
rules, such as a function arising from the application of the axiom of choice?
Wanting to allow such undescribable functions is the reason for the set
theoretic definition. I assume "rule" isn't a term of art?

What's the book to read for this stuff?

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ihm
Yes, program is a term of art. It means a "term" in the type theory under
consideration. A type theory is essentially a programming language with a
fancy type system, and there are many type theories, just as there are many
programming languages.

It does allow for functions without computable rules. In the case of the axiom
of choice, one postulates as an axiom a program with a type that says
something like

"give me a type A and an A-indexed family of types, P : A -> Type, such that
for all a:A, the type P a is non-empty, and I'll give you a function f : (a :
A) -> P a".

As far as I know, there is still no one comprehensive book on type theory. I
think the type theory parts of the homotopy type theory book
([https://hottheory.files.wordpress.com/2013/03/hott-
online-61...](https://hottheory.files.wordpress.com/2013/03/hott-
online-611-ga1a258c.pdf)) are quite good.

Chapter 1 gives the basics of dependent type theory.

Chapter 2 is about the groupoid interpretation of type theory and is quite
nice. Chapter 3 is about "logical things" in type theory (things like axiom of
choice).

Chapter 5 is about inductive types which are quite central to type theory
(these are generalizations of algebraic data types like trees and lists from
functional programming languages).

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transfire
Closely associated with the Navel Line of Sight.

