Here is the page that is confusing me: Page 25 of General Topology (Willard)
Definition 3.5 If $X$ is a topological space, and $E\subset X$, then the closure of $E$ in $X$ is the set $\overline{E} = \mathrm{Cl}(E) = \bigcap \{K\subset X\mid K\text{ is closed and }E\subset K\}.$
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Lemma 3.6 If $A\subset B$, then $\overline{A}\subset \overline{B}$.
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Theorem 3.7 The operation $A\mapsto \overline{A}$ in a topological space has the following properties:
K-a) $E\subset \overline{E}$
K-b) $\overline{(\overline{E})}=\overline{E}$
K-c) $\overline{A\cup B} =\overline{A}\cup\overline{B}$
K-d) $\overline{\varnothing} = \varnothing$
K-e) $E$ is closed in $X$ iff $\overline{E}=E$.
Moreover, given a set $X$ and a mapping $A\mapsto \overline{A}$ of $\mathscr{P}(X)$ into $\mathscr{P}(X)$ satisfying K-a through K-d, if we define closed sets using K-e, the result is a topology on $X$ whose closure operator is just the operation $A\mapsto\overline{A}$ we began with.
Everything was fine until this page, and suddenly I'm utterly confused. Definition 3.5 defines what a closure is, and lemma 3.6 follows directly from this definition and is, it seems to me, almost self-evident... and then Theorem 3.7 happens and I've no idea what's going on.
In the second-to-last paragraph, Willard implies that, in the collection F of all sets $\bar{A}=A$, $A \subset B \implies \bar{A} \subset \bar{B}$ does not directly follow from lemma 3.6.
We proceed now to the second part of the theorem. Let $X$ be any set and $A\to \overline{A}$ a mapping of $\mathscr{P}(X)$ into $\mathscr{P}(X)$ satisfying K-a through K-d. Let $\mathscr{F}$ be the collection of all sets $A$ such that $A=\overline{A}$. The assertion is that $\mathscr{F}$ satisfies F-a through F_c of Theorem 3.4.
First note that if $A\subset B$, then by K-c, $\overline{B}=\overline{A}\cup\overline{B-A}$, so that $\overline{A}\subset \overline{B}$ (why couldn't we just refer to Lemma 3.5?)
Wouldn't it follow even more clearly so? Isn't this valid: $ \bar{A} (= A) \subset \bar{B} (= B) \implies Lemma \; 3.6$
In fact, I'm having trouble understanding the significance of this Kuratowski closure operation altogether... don't K-a to K-e all follow from *Definition 3.5?
I think I might need a more extensive article on this topic, because I don't fully understand the significance of the material on this page.