In topologies is $\in$ the same as $\subset$?
Particularly, was thinking of the subspace topology:
Given a $(X, \tau)$ and $S \subset X$. A subspace top. is
$$\tau_S=\{ S \cap U : U \in \tau \}$$
Now since $\tau$ is some family of subsets of $X$, then it's a set.
Clearly $U$ is also a set, since it's intersected with a set.
But what then motivates saying $U \in \tau$, rather than $U \subset \tau$?
No. The topology $\tau$ is a set whose elements are sets contained in $X$. For an example, consider the topology
$$\tau := \{\emptyset, \{0\}, \{1\}, X\}$$
on the space $X = \{0, 1\}$. Then $\{0\} \in \tau$, but $\{0\} \nsubseteq \tau$ because $0 \notin \tau$.