Open Sets Explained

Open sets are fundamental to topology, defining the "openness" of spaces in a way that generalizes familiar notions from geometry.

Definition

A set \(U\) in a topological space is open if, for every point \(x \in U\), there exists a neighborhood of \(x\) fully contained in \(U\).

Examples

In \(\mathbb{R}\) with the standard topology, intervals like \((a, b)\) are open, but \([a, b]\) is not because endpoints lack full neighborhoods.

Role in Topology

Open sets form the basis of a topology, satisfying: (1) \(\emptyset\) and the space are open, (2) unions of open sets are open, (3) finite intersections are open.