I'm having trouble understanding topologies.
We say that $U \subseteq X$ is open if $U \in \tau$. If $(X, \tau)$ is a topological space and $U \subseteq X$, why are these properties the same?
- $U$ is open
- For each $x \in U$, there is an open $U_x$ with $x \in U$ and $U_x \subseteq U$.