Question

open cover of a set is denoted by \(\left\{ U_{\alpha} \right\}\). What does alpha stand for?

Answer 1

maybe an index?

Answer 2

indexing set i think @zzr0ck3r

Answer 3

can you give examples? like what does \(\left\{ U_{\pi} \right\}\)

Answer 4

...stand for?

Answer 5

um its just a stand in for an index

so the amount of sets you are taking the union of is \(\alpha\)

Answer 6

\(\pi\) would not make sense if we are talking about 3.14....

Answer 7

but we use pi for many things...

Answer 8

but alpha doesn't have to be a natural number tho?

Answer 9

no

Answer 10

you can have infinite and uncountable infinite

Answer 11

http://en.wikipedia.org/wiki/Cover_(topology)

Answer 12

Just really bad short notation....

Answer 13

if i remember correctly you could define alpha as an interval
\[\alpha_n = (1, n)\]

then the set \(\{U_{\alpha_{1}}\}\) refers to the set \((1, 1)\)

Answer 14

let's say the interval [0,1]. How would you denote that for every element in [0,1] take a ball of radius 1/10?

Answer 15

@zzr0ck3r please correct me if i talk nonsense :)

Answer 16

Yeah that seems fine to me. its just notation...

Answer 17

should be like this i think

then the set \(\{U_{\alpha_{1}}\}\) refers to the set \(\{(1, 1)\}\)

Answer 18

so you want the collection of balls around x s.t. x is in that set?

Answer 19

yes @zzr0ck3r

Answer 20

so is alpha is like the notation for sequence a(n) where a itself does not represent a number?

Answer 21

alpha is really bad notation for this

\(\cup_{i\in \alpha}A_i\)

Answer 22

where A_i is one of the setes in the collection

and there are \(\alpha\) sets

Answer 23

\(\{B(x,\frac{1}{10}) \mid x\in [0,1]\}\)

Answer 24

:O so alpha is a set?

Answer 25

yes, an indexed set

Answer 26

\(\alpha\) is a member of an index set, say \(\Gamma\)

Answer 27

\(\alpha\) is the indexed set

Answer 28

we would never just list a member of an index as subscript in a union, there would be no point

Answer 29

the whole time i thought it was a number! -.- that's why it got even more confused when it comes to the notation for subcover

Answer 30

every number is a set

;)

Answer 31

I mean in your example of your open cover \(\{U_\alpha\}\), \(\alpha\) is a member of some index set.

Answer 32

no, I think alpha is the indexed set, why would you union the elements of a set, you just get back the set.....

Answer 33

http://en.wikipedia.org/wiki/Cover_(topology)

Answer 34

second line

Answer 35

its bad notation lol

Answer 36

wikipedia is right, \(\alpha\) is a member of index set \(A\) there

Answer 37

but this short notation stands for the union of the indexed set.

Answer 38

read the next line

Answer 39

the next line is consistent, the collection of \(U_\alpha\) meaning \(\alpha\) runs through some index set. If you have learned topology before, it is a common notation that \(\alpha\) is a member of an index set.

Answer 40

For example you can write \(\{U_\alpha\mid \alpha \in [0,1]\}\) or \(\bigcup_{\alpha \in [0,1]}U_\alpha\)

Answer 41

I would never write it like this and this is why I say its bad notation. I would write it as

\(\cup_{I\in I}A_i\)

Answer 42

err \(\cup_{i\in I}A_i

\)

Answer 43

they wrote \(\cup_i\)

Answer 44

well, there is no different with your notation. They just don't specify, what is the exact index set, but that is ok, to save space and time.

Answer 45

I am saying what if we have sets \(A_1=\{1,2,3,4\},A_2=\{a,b,c,d\}\) and lets say our index is

then explin to me wht \(U_1\) is?

Answer 46

where our index is \(I=\{1,2\}\)

Answer 47

They don't specify the collection either...

Answer 48

so again this is just bad notation for what I said. It does not any longer stand for an element or we would just have \(\cup_1=\cup A_1=A_1\)

Answer 49

Yes, by you give a name to your set of index and one can avoid it either

Answer 50

Let say by saying \(\bigcup_\alpha A_\alpha\)

Answer 51

In this notation it is understood that you have an index set in the background where the \(\alpha\) is a particular member in this set.