### 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 is a topological space with a compatible metric , then the following metric is also compatible: for , .

**Propositioin **

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

**Examples **

- , ;
- The cantor space with the discrete topology in ;
- The Baire space with the discrete topology on ;
- The Hilbert cube , where .

**Lemma **If is a metric space, then closed sets are ; equivalently, open sets are .

**Proposition **A subset of a Polish space is Polish if and only if it is .

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

Considering first an open set , we exploit the fact that it does not contain its boundary point to define a compatible metric topology of that makes the boundary of “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 with a closed subset of by

where is a complete compatible metric for .

Let be completely metrizable and let be a complete compatible metric for . Define an open set as the union of all open sets that satisfy

- ,
- ,
- .

We show that .

### 2 Trees

For a nonempty set , we denote by the set of finite tuples of elements of , i.e.

where . For , we denote by the length of .

**Definition ** For a set , a subset of is called a (set theoretic) *tree* if it is cloded downward under .