Topological equivalence of function spaces

If $X$ is discrete then $C_p(X)$ is ${\mathbb R}^X$, which is not a very interesting observation. The question naturally arises how $C_p(X)$ is placed in ${\mathbb R}^X$ if $X$ is not discrete. It was shown in Lutzer and McCoy [20] that if $C_p(X)$ is a $G_\delta$-subset of ${\mathbb R}^X$ then $X$ is discrete. It was shown subsequently in Dijkstra, Grilliot, Lutzer and van Mill [14] that $C_p(X)$ cannot be an $F_\sigma$-subset or a $G_{\delta\sigma}$-subset of ${\mathbb R}^X$, unless $X$ is discrete (see also van Mill [26]). But it can be an $F_{\sigma\delta}$-subset of ${\mathbb R}^X$. This is true for example if $X$ is countable and metrizable.

For more results on the Borel complexity of function spaces, see e.g., Calbrix [10, 11], Lutzer, van Mill and Pol [21], Cauty [12], and Cauty, Dobrowolski and Marciszewski [13].

So the function spaces of familiar spaces as ${\mathbb Q}$ and $\omega+1$ all have the same Borel type. The question naturally arises whether the Borel type of $C_p(X)$ determines its topological type, or its linear-homeomorphism type. The examples due to Bessaga and Peczynski that we mentioned above, show that there are nonisomorphic function spaces of the same Borel type. So only the question about topological equivalence remains.

Let $Q$ denote the Hilbert cube $\prod_{n=1}^\infty [-1,1]_n$ with its so-called pseudo-boundary

$$B(Q) = \{x\in Q: (\exists n\in{\mathbb N})(\vert x_n\vert =1)\}.$$

The topological structure of function spaces of low Borel complexity is determined by the following result.

Theorem 1.
Let $X$ be a countable nondiscrete space such that $C_p(X)$ is an $F_{\sigma\delta}$-subset of ${\mathbb R}^X$. Then $C_p(X)$ is homeomorphic to the countable infinite product of copies of $B(Q)$.
As a consequence, $C_p({\mathbb Q})$ and $C_p(\omega+1)$ are homeomorphic.

Theorem 1. is due to Dobrowolski, Marciszewski and Mogilski [16] who proved it via the Bestvina-Mogilski [9] method of absorbing sets in the Hilbert cube. This is part of infinite-dimensional topology. A slightly stronger result is due independently to Baars, Gladdines and van Mill [5] and Dijkstra and Mogilski [15].

Since as we saw above nontrivial function spaces are at least of ` $F_{\sigma\delta}$ Borel complexity', Theorem 1. proves that at the first nontrivial stage all function spaces are homeomorphic. What happens at more complex situations is a mystery, although Cauty [12] proved that a result as elegant as Theorem 1. is not possible: there are countable spaces $X$ and $Y$ such that $C_p(X)$ and $C_p(Y)$ are not homeomorphic but have the same Borel type. The analysis of such more complicated function spaces is subject of current investigations.