Open set

An open set in a topological space $(X,\mathcal{T})$ is a subset $U\subseteq X$ such that $U\in\mathcal{T}$.

Open sets are the basic “observable” sets in topology: they define neighborhoods of points and determine operations like interior and closure . They also control continuity through preimages.

Examples:

• In $\mathbb{R}$ with the usual topology, $(0,1)$ is open.
• In the discrete topology on $X$, every subset of $X$ is open.
• In the indiscrete topology on $X$, the only open sets are $\varnothing$ and $X$.