/sci/ - Science & Math

[Anonymous No.6671523]

Is the open interval <0,1> isomorphic to [0,1]? I know that <0,1> is not compact (this can be shown as a direct result of Heine-Borel theorem) and [0,1] is compact. Is that sufficient to show that the two intervals are not isomorphic?

[Medical_Physicist !!ST92PtcX00D No.6671724]

Have you tried to establish a bijection between the two?

Protip: They're isomorphic.

[Medical_Physicist !!ST92PtcX00D No.6671727]

>>6671724
At least they are what I think you mean by isomorphic, insofar as there exists a bijection between them.

[Anonymous No.6671728]

>>6671523
Well fuck you, I'm trans on a sunny day, and cis when it gets hot.

[Anonymous No.6671883]

>>6671523
>Is that sufficient to show that the two intervals are not isomorphic?
if by "isomorphic" you mean "homeomorphism", yes it's sufficient.

If you're just interested by the bijection part, look at the pic where f is defined from [0;1] to ]0;1[


r_n is simply defined by:

N->E
n->r_n

E being the rational numbers which are in ]0;1[
(r_n is well defined since E is countable)


----
si=if in the pic, I'm a lazy boy

[Anonymous No.6671904]

>>6671523
Isomorphic as sets? Cantor's theorem says:
If there is an injection from <span class="math">A[/spoiler] to <span class="math">B[/spoiler] and there is an injection from <span class="math">B[/spoiler] to <span class="math">A[/spoiler], then there is a bijection between <span class="math">A[/spoiler] and <span class="math">B[/spoiler].

in this case:
<span class="math">\varphi\colon (0,1)\to [0,1],\quad x\mapsto x[/spoiler] is an injection from <span class="math">(0,1)[/spoiler] to <span class="math">[0,1][/spoiler], and
<span class="math">\psi\colon [0,1]\to(0,1),\quad x\mapsto \frac{1+x/3}[/spoiler] is an injection from <span class="math">[0,1][/spoiler] to <span class="math">(0,1)[/spoiler].

[Anonymous No.6671906]

>>6671904
...where by "Missing argument for \frac", I mean "<span class="math">\psi\colon [0,1]\to(0,1),\quad x\mapsto \frac{1+x}3[/spoiler]".

[Anonymous No.6671908]

Under the assumption that intervals have an order, you can use FO logic to prove that they're not isomorph

If you look at the structure A ([0,1], <) and B (]0,1[,<), then you can prove that they're not element equivalent with an FO-term:

for all x, there exists an y so that ( x < y or x = y)

Basically that there's a minimal element.

Because A and B aren't element equivalent then there cannot be an isomorphic relation

[Anonymous No.6671942]

>>6671908
>prove that they're not isomorph
but (as sets), they are.
You have two proofs just above

[Anonymous No.6671944]

>>6671523
Isomorphic in what category?
In Set, they are isomorphic. In Top, they are not. In HoTop, they are.

[Anonymous No.6673257]

medical physicists not even once

[Anonymous No.6674193]

>>6671904
I thought Cantor's theorem was about sets and their powersets...

[Anonymous No.6674202]

but that IS cis orientation...

[Anonymous No.6674537]

>>6674202
Yeah. It's someone telling that molecule to check its privilege. I guess.

[Anonymous No.6674755]

>>6671942
they mean as orders

[Anonymous No.6675392]

>>6674755
then they aren't, see >>6671908

[Anonymous No.6676022]

>>6671904
Can someone post a proof of: If there is an injection from A to B and there is an injection from B to A, then there is a bijection between A and B.
Or post a link to one/ what I should google to find a proof of it?