Definition talk:Filter on Set

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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.

A different school of thought suggests that it is better to avoid ambiguity whenever possible, even to the point of providing explanations of particular usage of a notation.

My view (and one which I have suggested as a house rule) is that every page should contain an explanation (or at least a link to one) of all potentially ambiguous notations on a page. I'm in a minority, though, so feel free to pay no attention, I'm always in trouble for being dogmatic. Real mathematicians have no time for such fussiness. --Prime.mover 07:25, 26 January 2010 (UTC)

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 $\emptyset$).
Again, there should be a standard for ProofWiki somewhere :) --Florian Brucker 11:41, 29 January 2010 (UTC)