A non-separable space is a topological space in which there does not exist a countable dense subset. In other words, there is no countable set of points that are "close" to every point in the space.
A separable space is a topological space in which there exists a countable dense subset. This means that there is a countable set of points that are "close" to every point in the space. In contrast, a non-separable space does not have this property.
Non-separable spaces are important in topology because they provide examples of spaces that have unique properties and cannot be easily studied using methods designed for separable spaces. They also have applications in mathematical analysis and functional analysis.
Yes, a non-separable space can be compact. Compactness is a property of a topological space that is independent of separability.
Yes, there are several real-life examples of non-separable spaces. These include the space of all continuous functions on a given interval, the space of all square-integrable functions on a given interval, and the space of all bounded sequences of real numbers.