Descriptive set theory

1 Polish space

Definition A topological space is called polish space if it is separable and completely metrizable (i.e. admits a complete compatible metric).

We work with Polish topological spaces as opposed to Polish metric spaces becasue we don’t want to fix a particular complete metric, we may change it to serve different purposes; all we care about is that such a complete compatible metric exists.

Lemma If X is a topological space with a compatible metric d, then  the following metric is also compatible: for x,y\in X, \overline{d}(x,y):=\min(d(x,y),1).

Propositioin 

  1. Completion of any separable metric space is Polish.
  2. A closed subset of a Polish space is Polish (with respect to relative topology).
  3. A countable disjoint union of Polish spaces is Polish.
  4. A countable product of Polish spaces is Polish (with respect to the product topology).

Examples 

  1. \mathbf{R}^{\mathbf{N}}, \mathbf{C}^{\mathbf{N}};
  2. The cantor space \mathcal{C}=2^{\mathbf{N}} with the discrete topology in 2;
  3. The Baire space \mathcal{N}=\mathbf{N}^{\mathbf{N}} with the discrete topology on \mathbf{N};
  4. The Hilbert cube I^{\mathbf{N}}, where I=[0,1].

 

Lemma If X is a metric space, then closed sets are G_\delta; equivalently, open sets are F_\sigma.

Proposition A subset of a Polish space is Polish if and only if it is G_\delta.

Proof Let X be a Polish space and let d_X be a complete compatible metric on X.

\Leftarrow Considering first an open set U\subset X, we exploit the fact that it does not contain its boundary point to define a compatible metric topology of U that makes the boundary of U “look like infinite” in order to prevent sequences that converge to the boundary from being Cauchy. In fact, instead of defining a metric  explicitly, we define a homeomorphism of U with a closed subset of X\times \mathbf{R} by

\displaystyle x\mapsto (x,\frac{1}{d_X(x,\partial U)}),

where d_X is a complete compatible metric for X.

\Rightarrow Let Y\subset X be  completely metrizable and let d_Y be a complete compatible metric for Y. Define an open set  V_n\subset X as the union of all open sets U\subset X that satisfy

  1. U\cap Y=\emptyset,
  2. \text{diam}_{d_X}(U)<1/n,
  3. \text{diam}_{d_Y}(U\cap Y)<1/n.

We show that Y=\bigcap_{n\in\mathbf{N}}V_n.

2 Trees

For a nonempty set A, we denote by A^{<\mathbf{N}}  the set of  finite tuples of elements of A, i.e.

\displaystyle A^{<\mathbf{N}}=\bigcup_{n\in\mathbf{N}}A^n,

where A^{0}=\{\emptyset\}. For s\in A^{<\mathbf{N}}, we denote by |s| the length of s.

Definition  For a set A, a subset T of A^{<\mathbf{N}} is called a (set theoretic) tree if it is cloded downward under \subset.