Question

teach me indexed sets @ikram002p

Answer 1

im not able to appreciate indexed sets on a set

Answer 2

you at least need to first understand partial orders.

Answer 3

Ohk, i can catchup i guess... i know about metric spaces and related stuff

Answer 4

indexed set is just an expression of sets that have something common related to another countable set .
ok so far ?

Answer 5

need not me countable

Answer 6

be*

Answer 7

the indexed set need not to be countable BUT it need to be defined with respect to countable sets , i'll give examples

Answer 8

okay so you're ordering the set based on index ?

Answer 9

as the most trivial example: \(\cup_{x\in \mathbb{R}}x=\mathbb{R}\)

Answer 10

x is the index here ?

Answer 11

this is not indexed :O , this is like union of countable infinite indexed sets

Answer 12

i thought you could only use natural numbers as index

Answer 13

u can use rational , natural or what ever it is hmm

Answer 14

http://math.stackexchange.com/questions/106102/use-of-sum-for-uncountable-indexing-set

Answer 15

any set with a total partial order can be an indexed set

Answer 16

what is partial order ?

Answer 17

http://en.wikipedia.org/wiki/Index_set

3rd example

Answer 18

its basically what you think of as \(\le\)

Answer 19

ok @zzr0ck3r teach him :)

Answer 20

google it.... read the one paragraph on wiki.

Answer 21

that looks like a bijection ?

Answer 22

looks you're pointing elements of one set from another set, based on that definition the index need not be a natural number i see

Answer 23

what does?

Answer 24

Its not function

Answer 25

tje link u gave has this

an index set is a set whose members label (or index) members of another set.[1] For instance, if the elements of a set A may be indexed or labeled by means of a set J, then J is an index set. The indexing consists of a surjective function from J onto A and the indexed collection is typically called an (indexed) family,

Answer 26

a partially ordered set, is a set A with a relation < where three properties need to occur

1) a<a (reflexive)
2)a<b and b<a implies a=b (anti symmetric)
3) a<b and b< c implies a<c (transitive)

one example would be the naturals with "divides"

Answer 27

ahh I thought you were talking about <

a surjection is an onto map, a bijection is a one to one and onto map

Answer 28

got it, and also got the definition of partially ordered set

Answer 29

think of a sequence, we use N to be the index.

but a subsequence of some sequence would have a subset of N as the domain, so we need not have our map be onto

Answer 30

Oh so a sequence definition also uses indexed sets is it ?

Answer 31

but the indexed set need not be countable. This shows up more in set theory/topology.

Answer 32

\[\{a_n\}\]

is \(n\) an index set of natural numbers ?

Answer 33

well its using an indexed set N to "index" the elements of the sequence

Answer 34

thanks i have an example if u have time.. the one you, me, chris and eliassaab have worked sometime back... one sec

Answer 35

the sequence itself could be thought of as an indexed set.

Answer 36

eliassaab is very strong in this topic. I would listen to what he said. I was not trying to cut you off @ikram002p I just saw someone telling this same user the same thing yesterday, so I wanted to make sure it was clear. I was just knocked off points for assuming my index set I was countable.

Answer 37

like 3 days ago, so its fresh:P

Answer 38

:D well i rather to learn instead of teaching :P
i made a mistake i guess by saying countable pardon me .

Answer 39

same here. On a damn test.

Answer 40

what does it mean by taking "union of an infinite indexed family in T" ?

\[\rm A \subset X , T=2^A \cup \{X\}\]

Answer 41

it means if all sets ( that follows some rule of indexed ) are in T
then show there union is also in T

Answer 42

family is just another word for sets, and it usually indicates that the elements of the set are themselves sets. They may use the word collection in the same manner.

Infinite means there are infinite of these elements (sets) in the family(set), and they are indexed by some set (T not sure without more). So if you pick an element in T, then it points to a set inside this infinite family.

Answer 43

Are they saying that T is the indexed set, or that T is a collection, and there is a subset of T of which we are to take the union of.

Answer 44

what do u mean by ?
. So if you pick an element in T, then it points to a set inside this infinite family.

Answer 45

If T is the indexed set

Answer 46

But I think I was reading it wrong

Answer 47

nope its not

Answer 48

I would need more context, but I would bet @ikram002p is right.

Answer 49

sorry let me rephrase it again :

i have a set defined as below
\[\rm T=2^A \cup \{X\}\]

where \(A \subset X \) and \(X\ne \varnothing\)

Answer 50

the point is , if there is a family of indexed set that all of its elements belong to T
then the union is also in T

Answer 51

and the question goes like this

`Show that union of an infinite indexed family in T is also in T.`

Answer 52

yep , its the third condition to define a topology

Answer 53

oh yeah what ^^ said

Answer 54

so should i first pick some set for indexing elements in T ?

Answer 55

so 2^A is the continuum cardinality after A?

Answer 56

2^A is powerset of A

Answer 57

\(\mathcal{P(A)}\)

Answer 58

that is not how we denote the power set of A

Answer 59

but it is how we denote the cardinality of the power set of A

Answer 60

well it only makes sense if its a set, so I guess I just have never seen this notation used to mean power set

Answer 61

ohk..

\[\mathcal{ T=P(A) \cup \{X\}}\]

Answer 62

dot pick up , just use the definition ,
\(\left \{ V_\alpha : \forall \alpha \in \Delta \right \}\in T \)

and using P(A) for power set is much better

Answer 63

so by this i picked any indexed set , which all of its elements belong to T , ok so far ?

Answer 64

I see \(\Delta \) is a set, what does it contain ?

Answer 65

I am not sure here, it screams Zorns lemma to me so I may be blinded. Ill watch and learn:)

Answer 66

im only looking for understanding indexed sets from analysis point of view

i don't think im ready for topology yet :O

Answer 67

This question actually seems quite trivial. Take an infinite union of elements in T, these elements are either subsets of A or they are all of X. If we take the union we will have two cases

1) X was one of our elements, then the union is X, and X is for sure an element in T

2) X was not one of our elements, then the union is a subset of A, so for sure its in T

Answer 68

(because the union of subsets of A, is a subset of A)

Answer 69

\[\large \mathcal {\bigcup_{i=1}^{\infty}U_i}\]

?

Answer 70

right, then either U_i = X for some i, or not. if it does....

Answer 71

Show that union of an infinite `"indexed family in T"` is also in T.

i don't get that phrase yet but ur solution makes sense!

Answer 72

well, I would say \(i\in I\)

\(U_{i\in I}u_i\) saying it like you did is assuming its countable, because we use that notation to mean indexed by the reals

Answer 73

err indexed by the naturals i mean

Answer 74

can i read it as

Show that `"infinite union of elements in T"` is also in T.

Answer 75

yes

Answer 76

-.- its not topology view lol
ok how u define a set that there elements are sets , and these sets (X_n) are intervals such that X_n =(0,1/n ) and n \in N
just make a neat notation thats all

Answer 77

who said anything about intervals?

Answer 78

yes you can read it like that @rsadhvika T is a "family" or "collection" because its elements are themselves sets

Answer 79

its just an example xD sorry if i mixed things up

Answer 80

idk what were talking about anymore , explaining the indexed or solving the question

Answer 81

ok so what I am I missing that the following does not work

Let \(\large \cup_{i\in I}x_i\) be a family of indexed sets, where \(I \) is an infinite indexed set.

case 1) \(X=x_i\) for some \(i\in I\) then \(\large \cup_{i\in I}x_i=X\in T\)

case 2) \(X\ne x_i \forall i\in I\) then \(\large \cup_{i\in I}x_i\subseteq A\) so \(\large \cup_{i\in I}x_i\in T\)

Answer 82

xD why T involved

Answer 83

ok lets forget the T example for a second , @rsadhvika u got what indexed family means ?

Answer 84

are u refering to

\[\large \cup_{i\in I}x_i\]

?

Answer 85

Why T involved????? The question defines T.

Answer 86

ok im completly ignored
nvm guys

Answer 87

lol I was just trying to solve the question that he posted from the other day. Not but in any more about indexed sets.

Answer 88

i think i get it, you're indexing the elements, \(x\)'s, using the elements from infinite set \(I\)

Answer 89

I will not but in any more*

Answer 90

yes, and remember the x's are sets.

Answer 91

they are either subsets of A or all of X

Answer 92

okay so there is a surjective mapping from I to T

Answer 93

why arent you using natural numbers as index set ?
natural numbers are also infinite so they can index infinite sets right ?

Answer 94

\(\large \cup_{i\in I}x_i\) is a family of indexed sets because each \(x\) is a subset. got it thanks :)

Answer 95

\(\large \cup_{i\in I}x_i\) is a family of oranges if each \(x\) is an orange

Answer 96

explicit definition of indexing set seems unnecessary but i get the general idea thanks to both of you!

Answer 97

here is an easy example where our indexed set is \(\mathbb{R}\)

Consider \[D:=\{\{x\} | \ x\in \mathbb{R}\}\]

This is a set of singletons (some call it a family or collection), for each \(x\in \mathbb{R}\) we have a set \(\{x\}\)

\(D\) is indexed by \(\mathbb{R}\).