Remark

$\mathbb{R}$ and $[0,1]$ are examples of Polish spaces.

Remark (Advantages to working with Polish Spaces)

• Precompactness can be defined using either Relatively Compact or Totally Bounded
• Allows for Prokhorov’s Theorem to work both ways
• $\mathbb{X}$ being polish means $\mathcal{P}(\mathbb{X})$ is polish.
• Presumably makes the process showing a space is Metrizable easier.
• Avoids many pathologies in probability theory.