# Definition talk:Filter on Set

Question: does the usage of $\subset$ in this definition denote a proper subset? That is, does $U \subset V$ imply $U \ne V$?
We've had discussions about this before: apparently topologists seem to resent the use of $\subseteq$ on the basis that anyone studying topology who can't adapt to the conventions don't deserve to.
Good point. I'm using $\subset$ and $\subseteq$ like I would use $<$ and $\le$. Thus $\subset$ indeed refers to a proper subset. Upon your remark I reread the definition and found that I should've used $\subseteq$ in two places instead of $\subset$. I corrected these. The remaining $\subset$ is fine, since the whole power set of a set is never a filter on that set (since it contains $\emptyset$).