What is the cardinal of the set of Uniform Continuous functions with domain in $\mathbb{R}$? I learned in my topology class that $\mathbb{R} \cong \mathfrak{C}(\mathbb{R})$
The latter being the set of continuous functions with domain in $\mathbb{R}$.
Since if a function is uniformly continuous then it is continuous, the set of all uniform function must have cardinal $\leq \mathfrak{c}$.
But continuous functions need not be uniformly continuous; does this show that there is no bijection from continuous functions to uniformly continuous functions?
I'm kind of lost with all this cardinality stuff; it's not intuitive at all.
 A: Two sets $A$ and $B$ are said to have the same cardinality, written $|A| = |B|$, if there is a way to pair up each element of $A$ uniquely with an element of $B$ in such a way that you have no elements left over. In other words, $|A|=|B|$ if and only if there exists a bijection $f: A \to B$.
We can also define what it means to say $|A| \leq |B|$, with our motivation coming from the above idea. Instead of requiring that our pairing process exhausts all of the elements of $B$, we simply require that every element of $A$ be uniquely paired with an element of $B$. In other words, $|A| \leq |B|$ if and only if there exists an injection $f: A \to B$. Perhaps worth mentioning is that an equivalent condition is that there is a surjection $g: B \to A$.
As an example, the set of even integers $2\mathbb{Z}$ has the same cardinality as $\mathbb{Z}$, as the function $f: \mathbb{Z} \to 2\mathbb{Z}$ given by $f(x) = 2x$ is a bijection. Note that this shows that a proper subset doesn't necessarily need to be 'smaller'.
Let
$$ S = \{ f \in C(\mathbb{R}) \mid \text{$f$ is uniformly continuous}\}$$
As you've noted, $C(\mathbb{R})$ has cardinality $\mathfrak{c} = |\mathbb{R}|$, so we know that $|S| \leq \mathfrak{c}$ since $S$ is a subset of $C(\mathbb{R})$ (and hence the inclusion map shows that $|S| \leq |C(\mathbb{R})| = \mathfrak{c}$).
On the other hand, $\mathfrak{c} \leq |S|$, since for each $a \in \mathbb{R}$ the function $f_a : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = ax$ is uniformly continuous, which provides an injection $F: \mathbb{R} \to S$ given by $F(a) = f_a$.
The implication that $|S| \leq \mathfrak{c}$ and $\mathfrak{c} \leq |S|$ implies $\mathfrak{c} = |S|$ follows from the Cantor-Schroeder-Bernstein Theorem.
