Compactness

In mathematics, compactness is a property of a space that intuitively means it is limited and contains all its boundary points, similar to a closed and bounded shape. It ensures that any sequence of points within the space has a subsequence that converges to a point also in the space. This property is important because it guarantees certain behaviors, like functions achieving maximum values, and makes analysis easier. Think of a compact space as a well-contained, "neat" set where everything is contained within bounds, allowing for more predictable and manageable mathematical analysis.