Can R be an Indexing set?

Question

I was questioning my teacher about index sets and family of sets when he said " we could take our index set to be R also". Now from what i know( i am new to set theory and am studying Family of sets), i use indexing to arrange/ number the sets in a given collection( basically creating an one to one correspondence between the index set and the collection).So far i have encountered the set of natural numbers N being used as an index set for finite and infinite family of sets. My question is how can i use real numbers set R as an index set when it contains negative numbers? what about the negatives?

Answer 1

Any non-empty set can be an indexing set. You can for instance, define, for every $t\in\Bbb R$, $I_t=(-\infty,t)$. There is no problem with that.

Answer 2

Here's a thought experiment. While thinking about it consider what you intuitively and linguistically consider the word "index" to mean.

Suppose we want to index all the points of a circle. That if a circle is centered at $(2,3)$ and has radius $4$, and we want to index all the points $(x,y)$ of the circle based what the interior angle is. So $p_0 = (6,3)$ which is the first point with an interior angle of $0$. And $p_{\pi} = (-2,3)$ is the point where the angle is $\pi$ and $p_{\frac \pi 2} = (2,7)$ and $p_x = (2+4\cos x, 3+4\sin x)$.

That's an uncountable number points all indexed based an a criteria. Is that an index? Why or why not? We haven't (and can't) index the points into a list where we can list them all one after another. But you have indexed them so that we can "look up" any point based on its angle.

Is that what "index" means?[1]

Is it an index any different than a function? Either in any practical sense or usage.[2]

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[1] in my opinion, yes.

[2] in my opinion, there is no practical difference-- they are both a one way mapping of a set to another set but usage an index is specifically for reference and referral; a function is flexible in its purposes and goals.

oh, I suppose an argument can be made as the points of a set are distinct that an index must be an injective map... and technically a function is a collection of the input/output pairs and a value's "index" refers specifically only to the input referrer. So they are different I suppose.

But a for any indexed set $B$ then $f:(indices\ of\ B) \to B$ via $i \mapsto b_i$ is a function. ANd for any bijective function $f:A \to B$ then setting $b_i = f(i)$ is an indexing of the set $B$.

Yeah.... that's my answer and I'm sticking to it.

Answer 3

Here is a concrete simple example. For every $r\in\mathbb{R}$, define a family of functions $(f_r:\mathbb{R}\to\mathbb{R})_{r\in\mathbb{R}}$ by $f_r(x)=x+r$