## On The Cardinality of Function Spaces (Pt. I Real Functions)

**Point of post: **This post will be one of a few that I will post in my free time about the cardinality of certain function spaces. Thus, for example in this post we will show that

without using cardinal arithmetic, just basic injection, bijection stuff. From here we will proceed with a few more specific examples, ending with a general result that bounds the cardinality of (space of continuous functions) for a Hausdorff space .

*Motivation*

Often in mathematics one can ask questions which, at first seem unanswerable. For example, ask a freshman analysis student what the cardinality of and it looks practically impossible. That said, often interesting questions such as these have answers, just not one’s apparent at first glance. In this post we will start out easy and find the cardinality of for a given infinite set , a corollary of which will state that:

*Remark: *This is meant just to be informative, and fun. That said, I do not consider deeply set-theoretic topics, in particular I shall speak freely of sets without mentioning their universe and I will not explicitly mention the axiom of choice, etc. when invoked.

**Theorem: **Let be a set with . Then, .

**Proof: **Note that since it suffices to prove that . To do this, we first note that we may evidently define

where, as usual, is the indicator function given by

This is evidently injective since if then we may assume without loss of generality that and so choosing we see that so that . Conversely, we recall the basic fact that . In particular, since we know that . With this said, we consider the function

where , the graph of . This is also evidently injective since if then for some and so but . Thus, it follows that if is the guaranteed bijection that is an injection.

It follows from the Schroeder-Bernstein theorem as desired.

*Remark: *Note that this implies that where is the third Beth number.

**References:**

1. Jech, Thomas J. *Set Theory*. Berlin: Springer, 2003. Print.

basically you are saying it is \aleph_2 = | {\mathbf{R} }^{\mathbf{R}}|

Which is fine, however, can you show that this F is dense in recursive functions?

Comment by Noga | November 21, 2011 |

Noga,

I’m not even sure what that means haha. Please elaborate!

Best,

Alex

Comment by Alex Youcis | November 22, 2011 |