Open Set

Definition

$U\in \mathcal{T}$

Consider a Topological Space $(X,\mathcal{T})$. The elements of the Topology $\mathcal{T}$ are called the open sets of $X$

Definition in Metric Topology

Consider a Metric Space $(X,{\mathcal{T}}_d,d)$ $U\in {\mathcal{T}}_d⟺\forall y\in U,\exists \delta >0\text{s.t.}{B}_d(y,\delta )\subset U$

Facts

Every point of $U\in \mathcal{T}$ is an interior point of $U$.

Consider a Topological Space $(X,\mathcal{T})$ and a subset $U\subset X$. Then, $U$ is open in $X$ if and only if for an arbitrary point $x$ in $U$, there exists an open subset $U_x\subset U$ containing $x$. $U\in \mathcal{T}⟺\forall x\in U,\exists U_x\ni x\text{s.t.}{U}_x\in \mathcal{T}\text{and}{U}_x\subset U$

$A$ is an open set if and only if it’s equal to its Interior $A=\mathrm{int}A$.