> [!definition] > > Let $(X, \topo)$ be a [[Topological Space|topological space]]. The elements of $\topo$ are called **open sets**. > [!theorem] > > A set $A$ is open if and only if it is a [[Neighbourhood|neighbourhood]] of itself. > > Alternatively phrased, it is only open if it contains none of its [[Boundary|boundary]]. > > *Proof*. Suppose that $A$ is open, then $A$ is an open set containing itself and therefore a neighbourhood. > > Suppose that $A$ is a neighbourhood of itself, then it's contained by its [[Interior|interior]]: $A \subset A^o$, $A = A^o$ and $A$ is open. > [!theorem] > > Let $(X, \topo)$ be a topological space and $A \in \topo$, then a set $B \subset A$ is open in the [[Relative Topology|induced topology]] $(A, \topo_A)$ if and only if $B \in \topo$.