It appears that some topologists (in particular those who I am learning from) really like nhoods but not so much open sets. I say this because their definitions always involve nhoods where they could equivalently (and seemingly more intuitively) involve just open sets.
Here I take a nhood of $x\in (X,\tau)$ to be any subset of $X$ containing an open set that contains $x$.
Definition: $X,Y$ are two topological spaces. A function $f:X\rightarrow Y$ is continous at $x\in X$ iff for every nhood $V$ of $f(x)$ there is a nhood $U$ of $x$ such that $f(U)\subseteq V$.
But an equivalent definition arises when we replace 'nhood of' with 'open set containing' (i.e. instead of nhood of $x$ we say open set containing $x$). This seems much more natural to me, which leads me to ask: am I missing something here? Why/is it better to use nhoods in definitions instead of open sets.
To me nhoods seems rather clumsy compared to open sets.