Open Sets Definition I understand that the definition of an open set in a Metric Space and this can transfer over if you're dealing with a Metric Topological Space. However, I'm not sure if there is a standard definition of an open set in a Topological Space. I have read that it may vary under constraints.
I found one definition on Wikipedia:
If ($X$, $D$) is a Topological Space, then a subset $O$ of $X$ is open if and only if $O$ is a neighbourhood of each of its points.
This is how I expressed it mathematically:
for all x $\in$ O there exist an open set U containing x such that U  $\subseteq\ O $

Given a topology $\tau$ on a set $X$, a set $\mathcal{O}$ is called open if and only if $\mathcal{O} \in \tau$. A topology is a collection of subsets of $X$ such that it contains $X$, is closed under all unions, and closed under finite intersections. This definition is consistent with the metric space definition of open.
The Wikipedia article on open sets gives a more in-depth explanation. The general topology article is another good place to start.
