Polish Space
Definition
A topological space $X$ is called a Polish space if it satisfies the following conditions:
- (i): $X$ is a metrizable space.
- (ii): $X$ is a separable space.
- (iii): $X$ is a complete space.
Explanation
The term Polish Space directly comes from ‘Poland’, as you might guess from the naturalized expression. The reason behind this naming is that this concept was first actively researched by topologists and logicians from Poland. To delve a bit into the name, if in condition (i), instead of being just a metrizable space, it is exactly a metric space $(X,d)$, it is called a Polish metric space. However, in the actual applications of Polish spaces, the condition of being metrizable is often considered too general, and if it effectively suffices to be a metric space, it is commonly referred to simply as a Polish (metric) space instead of a Polish space. Polish spaces add the property of separability to the conventional space of a complete metric space. Being separable is closely related to being countable, making it especially applicable in measure theory, and notably in probability theory.