Polish spaces

A Polish space is a type of mathematical space that combines two key features: it is complete (meaning all limits of sequences within it stay inside the space), and it has a countable dense subset (meaning there are points so plentiful that any point in the space can be approximated arbitrarily closely by these special points). These spaces are important in areas like analysis and topology because they provide a flexible yet well-behaved setting for studying limits, continuity, and convergence. Examples include the real numbers with their usual distance and many function spaces used in mathematics.