Mathematical spaces are abstract concepts used to describe collections of points and the relationships between them. They provide a framework for understanding various mathematical structures, such as Euclidean spaces, which deal with familiar geometric shapes, and metric spaces, which focus on distances between points. These spaces can be finite or infinite and can have different dimensions. For example, a vector space consists of vectors that can be added together and multiplied by scalars. Understanding mathematical spaces is essential for fields like topology, analysis, and geometry, as they help mathematicians explore complex ideas in a structured way.