Tweak the definition of filter in textbook Introduction to Set Theory 3rd by Hrbacek and Jech

Question

My textbook Introduction to Set Theory 3rd by Hrbacek and Jech defines filter as follows:

1.1 Definition Let $S$ be a nonempty set. A filter on $S$ is a collection $\mathcal F$ of subsets of $S$ that satisfies the following conditions:

(a) $S \in \mathcal F$ and $\emptyset \notin \mathcal F$.

(b) If $X \in \mathcal F$ and $Y \in \mathcal F$, then $X \cap Y \in \mathcal F$.

(c) If $X \in \mathcal F$ and $X \subseteq Y \subseteq S$, then $Y \in \mathcal F$.

I found that we can omit the criteria $S \in \mathcal F$ by adding another one $\mathcal F \neq \emptyset$.

• $S \in \mathcal F \implies \mathcal F \neq \emptyset$

• $\mathcal F \neq \emptyset \implies \exists X \in \mathcal F$. On the other hand, $X \subseteq S$. Then $S \in \mathcal F$.

Please confirm if my reasoning is correct or not! Thank you so much!

Answer 1

You're correct of course. You could reformulate (without (a)) as:

"A filter on $S$ is a non-empty collection $\mathcal{F}$ from $\mathscr{P}(S)\setminus \{\emptyset\}$ such that (b) and (c) hold.

But that sort of "hides" a point and didactically they maybe wanted to stress explicitly that $S \in \mathcal{F}$ and also $\emptyset \notin \mathcal{F}$; it'a textbook after all, not a treatise striving for minimal definitions. It's more "concrete" in the way the state it. Also it hints to the minimal filter on $S$: $\{S\}$.