In [[topology]], an open set is a fundamental concept that helps define the structure of a topological space. Here’s a breakdown of what an open set is: 1. **Definition of Open Set**: An open set in a topological space is a set of points that, intuitively, does not include its boundary. More formally, a set $U$ within a topological space $X$ is considered open if, for every point $x$ in $U$, there exists a neighborhood of $x$ that is entirely contained within $U$. This neighborhood can vary in size and shape depending on the topology defined on $X$, but it must contain xxx and fit completely inside $U$. 2. **Neighborhood Concept**: A neighborhood of a point $x$ in a topological space is a set that includes an open set containing $x$. The requirement that every point in an open set has a neighborhood entirely within the set itself is central to many topological definitions and theorems. 3. **Examples**: - In the standard topology on the real numbers $\mathbb{R}$, an open interval such as $(a, b)$ (where $a < b$) is an open set. This is because, for any point xxx in $(a, b)$, you can find a smaller interval $(x-\epsilon, x+\epsilon)$ that is still within $(a, b)$ for some $\epsilon > 0$. - In a discrete topology, where every subset is defined as open, even singleton sets are open. - Conversely, in an indiscrete topology (the trivial topology), the only open sets are the empty set and the entire space itself. 4. **Importance in Topology**: Open sets are crucial in topology because they define continuity, convergence, and the overall topological structure of spaces. A function between two topological spaces is continuous if the preimage of every open set is open. This highlights how topologies can be compared and studied through the behavior of functions. Understanding open sets provides a base for exploring more complex topological concepts such as closed sets, compactness, connectedness, and more, each of which relies on the foundational idea of openness in a space. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Open Set") sort title, authors, modified, desc ```