How can one the cardinality of function space $C(X, Y)$ equipped with uniform topology, where $X$ is a Tychonoff space and $(Y, d)$ a metric space.

Question

How can one the cardinality of function space $C(X, Y)$ equipped with uniform topology, where $X$ is a Tychonoff space and $(Y, d)$ a metric space.

I mean how does the cardinality of $X$ and $(Y, d)$ help in finding the cardinality of function space $C_u(X, Y)$.

Answer 1

$|C(X,Y)|$ will depend on the combination of the topologies on $X$ and $Y$ and $|X|$ and $|Y|$ as well.

Some examples: if $X$ is $\Bbb R$ in the usual topology and $Y = \Bbb Q$, in the usual metric, then $|C(X,Y)| = \aleph_0$, because only the constant functions from $X$ to $Y$ are continuous. We also have the trivial bound $|C(X,Y)| \ge |Y|$ because we always have those constant functions in the set of continuous functions.

If $X=\Bbb R$ in the discrete topology (indeed Tychonoff) and $Y$ is any metric space, then all $f: X \to Y$ are continuous, so $|C(X,Y)| = |Y|^{|X|}$ so when $Y=\Bbb Q$ this equals $\aleph_0^\mathfrak{c} = 2^\mathfrak{c}$, e.g.

So the size can differ quite dramatically, and has to be considered on a case by case basis.