Can someone demonstrate a concrete example of when to use $\in$ and $\subseteq$ when talking about topologies and bases?
When is something $\subset$ of a basis or a topology and when is something $\in$ basis or topology?
I am learning topology and getting super confused with (sometimes) seemingly arbitrary interchangeability of $\in$ and $\subseteq$ which causes me to have trouble with basic proofs.
For example, in the definition of a topology generated by a basis:
$$\tau_\mathcal{B} = \{U \subseteq X| \forall x \in U, \exists B \in \mathcal{B} \text{ s.t. } x \in B \subseteq U\}$$
Let $B_x$ be the basis element associated with $x \in U$, then is: $$\bigcup_{x \in U} B_x \in \mathcal{B} \text{ or } \bigcup_{x \in U} B_x \subseteq \mathcal{B}?$$
Also
should we write $\mathcal{B} \in \tau_\mathcal{B}$ or $\mathcal{B} \subset \tau_\mathcal{B}$?
In hoping to see this more clearly, let's start with an example. Consider $\tau$ the euclidean topology on $\mathbb{R}$, that is to say the topology induced by the distant $d(x,y) := |x-y|$.
The open interval $(1,2)$ is an element of the topology $\tau$ for any, i.e. $(1,2)$ is open. Therefore $(1,2) \in \tau$. More generally, any interval of the form $(n,n+1)$ is in $\tau$. That is to say $(n,n+1) \in \tau$ for any fixed $n \in \mathbb{N}$. Therefore $$ \big\{ (n,n+1) :~~ n \in \mathbb{N}\big\} \subseteq \tau$$ since the set on the left is a collection of open sets.
Now you probably know that any countable union of open sets is open. Therefore $\bigcup\limits_{n \in \mathbb{N}}(n, n+1) \in \tau$.
To sum up $\big\{ (n,n+1) :~~ n \in \mathbb{N}\big\}$ is a subset of $\tau$ since it is a collection of open sets whereas $\bigcup\limits_{n \in \mathbb{N}}(n, n+1)$ is an element of $\tau$ since it is an open set.