THIS CONTINUES MY ANSWER THAT BEGINS HERE

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4. Union-Preserving Closure Operators $\;$ [satisfy (K1)] ==

Definition: We say that $T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$ is a union-preserving closure operator on $X$ if $T$ is a monotone closure operator on $X$ such that for each $A,\,B \in {\mathcal P}(X)$ we have $T(A \cup B) = T(A) \cup T(B).$

The term additive operator is often used, but I thought “additive” creates too much mental interference with unintended meanings from other areas of mathematics. Also, closure spaces arising from closure operators that are only required to satisfy (K1) do not seem to be of enough interest for the use of “union-preserving” here to cause problems. An example of a monotone closure operator that is not a union-preserving closure operator is the interior operator on $X = {\mathbb R}.$ Note that the interior operator satisfies (semi-K1) (and also (K2) and (K4), for those keeping track), but not (K2) (consider $A = \mathbb Q$ and $B = {\mathbb R} – {\mathbb Q}).$

I do not know of a naturally occurring example of a union-preserving closure operator that doesn’t also satisfy at least one of the remaining axioms for a topological operator, let alone an example from which useful insight is provided by considering it in the context of a closure operator. I also do not know how to obtain a “minimally differing” union-preserving closure operator from a monotone closure operator that doesn't explicitly make use of existing union-preserving closure operators, but the following at least shows that there exists a greatest union-preserving closure operator that is less than or equal to a given monotone closure operator. In fact, the same method works for an unrestricted closure operator $-$ see Hammer [21] (definition of $f_6$ at the bottom of p. 70).

Union-Preserving Modification of a Monotone Closure Operator: Given a monotone closure operator $T:{\mathcal P}(X) \rightarrow {\mathcal P}(X),$ the greatest union-preserving closure operator $T^{\text {u-p}}$ on $X$ that is less than or equal to $T$ can be defined by letting

$$ T^{\text {u-p}}(A) \; = \; \bigcup\,\{T'(A): \; T' \leq T \; {\text {and}} \; T' \; {\text {is a union-preserving closure operator on}} \; X \}. $$

Regarding the assumption (K1), the following may be of interest:

The classical topologists have persisted in requiring additivity of the closure function in the definition of topological spaces. From one standpoint the additivity axiom might be called the sterility axiom. That is to say, it requires that two sets cannot produce anything (a limit point) by union that one of them alone cannot produce. On the other hand, of course, as with the analogous independence of events in probability theory, the known presence of additivity produces special benefits. (from Hammer [21], top of p. 65)

5. Base Closure Operators $\;$ [satisfy (K1), (K2)] ==

Definition: We say that $T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$ is a base closure operator on $X$ if $T$ is a union-preserving closure operator such that $T(\emptyset) = \emptyset.$

Sometimes (but not here) the requirement $T(X) = X$ is included (e.g. see bottom of p. 4700 of Bliedtner/Loeb [6]). I believe the name originated in Lukeš/Malý/Zajíček [36], but I don’t know why they used the specific word “base”. A rather trivial example of a union-preserving closure operator that is not a base closure operator is given by $T(A) = X$ for all $A \in {\mathcal P}(X),$ with $X \neq \emptyset.$ Most of the results I give here concerning base closure operators, and many other results not included here, can be found in Chapter 1: Base Operator Spaces (pp. 5-37) of Lukeš/Malý/Zajíček [36].

Base Modification of a Union-Preserving Closure Operator: Given a union-preserving closure operator $T:{\mathcal P}(X) \rightarrow {\mathcal P}(X),$ we can obtain a base closure operator $T^{\text {base}}$ on $X$ by redefining $T$ to have the value $\emptyset$ at $A = \emptyset:$

$$ T^{\text {base}}(A) \; \triangleq \; \left\{\begin{array}{c} \emptyset & \text{if} & A = \emptyset \\ T(A) & \text{if} & A \neq \emptyset \end{array} \right. $$

Note that $T^{\text {base}}$ is the greatest base closure operator less than or equal to $T.$

The prototypical example of a base closure operator is the derived set operator of a topological space:

Example (derived set operator): Let $(X,\tau)$ be a topological space and define $T$ by

$$ T(A) \; = \; \{x \in X: \; A \cap (U - \{x\}) \neq \emptyset \;\; \text{for each} \; \tau \text{-open set} \; U \; \text{containing} \; x\}. $$

Then $T$ is a base closure operator on $X.$

By replacing “$A \cap (U - \{x\}) \neq \emptyset$” with “$A \cap U$ is not small” for various notions of “not small” (i.e. by using various stronger notions of being a limit point of a set), we obtain many other examples of base closure operators. For example, in the appropriate spaces we could require $A \cap U$ to not be finite, not be countable, not be a first Baire category set, not have Lebesgue measure zero, etc. In fact, the motivation in Lukeš/Malý/Zajíček [36] for considering the abstract notion of a base closure operator was to generalize all the various notions in potential theory for a set to be thin/not-thin at a point. Indeed, if $T$ is a base closure operator on $X$ and $x \in X,$ then $\{A \in {\mathcal P}(X): \; x \notin T(A) \}$ is an ideal of sets that we could call the $T$-thin sets at $x.$ Moreover, if $X$ is a set and if for each $x \in X$ we specify a nonempty ideal ${\mathcal I}_x$ of subsets of $X,$ then there exists a base closure operator on $X$ such that, for each $x \in X,$ we have ${\mathcal I}_x$ equal to the collection of $T$-thin sets at $x.$ Regarding the last two sentences, see Lukeš/Malý/Zajíček [36] (1.A.2 on p. 10 and 1.A.17 on p. 16).

Definition ($T$-topology): Let $T$ be a base closure operator on $X$ and let $C \in {\mathcal P}(X).$ We say that $C$ is a $T$-closed set if $T(C) \subseteq C.$

It is not difficult to show, for a fixed base closure operator $T,$ that the collection of $T$-closed sets satisfies the axioms to be the closed sets for some topology ${\tau}^T$ on $X.$ Of course, the corresponding topological closure operator, which I will denote by ${\tau}^T{\text {-}}{\text {Cl}},$ might not be equal to $T,$ but we always have $\;T \leq {\tau}^T{\text {-}}{\text {Cl}}.$ [[ Proof of this last inequality: Choose $A \in {\mathcal P}(X)$ and let $C$ be any $T$-closed set containing $A.$ Since $A \subseteq C,$ we have $T(A) \subseteq T(C);$ and since $C$ is $T$-closed, we have $T(C) \subseteq C.$ Therefore, $T(A)$ is a subset of any such set $C,$ and hence $T(A)$ is a subset of the intersection of all such sets $C.$ Thus, for each $A \in {\mathcal P}(X)$ we have shown that $T(A) \subseteq {\tau}^T{\text {-}}{\text {Cl}}(A).$ ]]

If $(X,\tau)$ is a topological space, then of course the topological closure operator ${\tau}{\text {-}}{\text {Cl}}$ for $(X,\tau)$ is a base closure operator on $X.$ Moreover, as is easy to check and what one would expect, the topological closure operator corresponding to ${\tau}{\text {-}}{\text {Cl}}$ (regarding ${\tau}{\text {-}}{\text {Cl}}$ as a base closure operator) is equal to ${\tau}{\text {-}}{\text {Cl}}.$ That is, we have ${\tau}^{({\tau}{\text {-}}{\text {Cl}})}{\text {-}}{\text {Cl}} = {\tau}{\text {-}}{\text {Cl}}.$

The topologies that arise in this way in potential theory $-$ the topologies ${\tau}^T$ where $T$ is a base closure operator, defined using a limit point notion in which a set is NOT $T$-thin at a point $x$ $-$ are called fine topologies (Wikipedia page and google search) because, in Euclidean spaces, these topologies are finer (i.e. have more open sets) than the usual Euclidean topology. [Note that a more stringent condition to be a limit point causes fewer points to be added to the set to get its topological closure, hence the topological closure is smaller, hence more closed sets are needed to obtain (via intersection) this smaller topological closure, hence there are more open sets. (There is probably a less circuitous way of seeing this.)] An alternative way that fine topologies like this arise in potential theory is that it is useful to consider topologies on ${\mathbb R}^n$ (or more general spaces) such that, for various collections of functions from ${\mathbb R}^n$ to $\mathbb R$ that are larger than the collection of ordinary continuous functions, all functions in the larger collection will be continuous. This is accomplished by using finer topologies on ${\mathbb R}^n$ and the usual topology on ${\mathbb R}.$

6. Čech Closure Operators $\;$ [satisfy (K1), (K2), (K3)] ==

Definition: We say that $T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$ is a Čech closure operator on $X$ if $T$ is a base closure operator on $X$ such that for each $A \in {\mathcal P}(X)$ we have $A \subseteq T(A).$

Note that a Čech closure operator differs from a topological closure operator in that (K4), which says that $T = T^{2},$ is weakened to $T \leq T^{2}.$ Thus, if a Čech closure operator $T$ satisfies $T^2 \leq T,$ then $T$ is a topological closure operator.

The corresponding generalized topological spaces are also called pretopological spaces. I’m using Čech’s name because Eduard Čech (1893-1960) studied these generalized closure functions extensively in his massive 893 page book [9] (1966 English translation). Čech also published a survey study of these notions in 1937. For some historical details about this part of Čech’s work, see Koutsk [34] (especially pp. 108-109). For instance, Čech was motivated by the fact that the idempotency property of closure was not satisfied by many spaces of functions for various notions of convergence. “He supposed that this transition to the more general concept would create no difficulties for the participants of the seminar and therefore without any hesitation he went on with his lectures. However, there were some difficulties. The concept of closures of sets supposed by axiom (4) [= idempotency] had been meanwhile so deeply rooted in the considerations of some members that misunderstandings occurred too frequently.” (Koutsk [34], near bottom of p. 108)

An example of a base closure operator that is not a Čech closure operator is the ordinary derived set operator $D:{\mathcal P}(\mathbb {R}) \rightarrow {\mathcal P}(\mathbb {R}),$ where for example $D(\mathbb Z) = \emptyset,$ and hence $A \subseteq D(A)$ is false for $A = {\mathbb Z}.$

Example (zero distance from a set): Let $X$ be a set and let $d:X \times X \rightarrow [0,\infty)$ satisfy $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,\,y \in X.$ Thus, $(X,d)$ is what we get if we begin with a metric space and drop the triangle inequality and drop the assumption that $d(x,y) = 0$ implies $x = y.$ Čech [9] calls this a semi-pseudometric space. Then $T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$ is a Čech closure operator, where $T$ is defined by

$$ T(A) \; = \; \left\{x \in X: \; \inf_{a \in A} d(x,a) \leq 0 \right\}. $$

The reason for using $\leq 0$ rather than $= 0$ is to accommodate the convention that $\inf \emptyset = -\infty.$ Čech [9] (p. 302) observes that if ${\lambda}^{*}$ and ${\lambda}_{*}$ are the Lebesgue outer and inner measures on $[0,1],$ then $d^{*}(A,B) = {\lambda}^{*}(A\,\triangle\,B)$ defines a pseudometric on ${\mathcal P}([0,1])$ and $d_{*}(A,B) = {\lambda}_{*}(A\,\triangle\,B)$ defines a semi-pseudometric on ${\mathcal P}([0,1])$ that is not a pseudometric on ${\mathcal P}([0,1]).$ $(A\,\triangle\,B$ is the symmetric difference of $A$ and $B.)$

Čech Modification of a Base Closure Operator: Given a base closure operator $T:{\mathcal P}(X) \rightarrow {\mathcal P}(X),$ we can obtain a Čech closure operator $T^{\text {Čech}}$ on $X$ by letting $T^{\text {Čech}}(A) \triangleq A \cup T(A)$ for each $A \in {\mathcal P}(X).$

It is not difficult to show that $T^{\text {Čech}}$ is the smallest Čech closure operator greater than or equal to $T.$

Since every Čech closure operator is a base closure operator, the same notion of a $T$-closed set continues to apply when $T$ is a Čech closure operator and, for a fixed Čech closure operator $T,$ the collection of $T$-closed sets satisfies the axioms to be the closed sets for some topology ${\tau}^T$ on $X.$ Thus, repeating what I’ve already said in the more general case of base closure operators, for any Čech closure operator $T$ we have $\;T \leq {\tau}^T{\text {-}}{\text {Cl}},$ where ${\tau}^T{\text {-}}{\text {Cl}}$ is the topological closure operator for the topological space $(X,{\tau}^T).$

Now suppose we begin with a base closure operator $T.$ From $T$ we obtain the Čech modification $T^{\text {Čech}},$ and from $T^{\text {Čech}}$ we obtain the collection of $T^{\text {Čech}}$-closed sets, and from the collection of $T^{\text {Čech}}$-closed sets we obtain the topological closure operator ${\tau}^{T^{\text {Čech}}}{\text {-}}{\text {Cl}}$ for the topological space $(X,{\tau}^{T^{\text {Čech}}}).$ Since the collection of $T$-closed sets is identical to the collection of $T^{\text {Čech}}$-closed sets (proof in a moment), it follows that $(X,{\tau}^T) = (X,{\tau}^{T^{\text {Čech}}}).$ [[ Proof: $(\subseteq)$ Let $A \subseteq X$ be $T$-closed. Then $T(A) \subseteq A.$ Hence, $T^{\text {Čech}}(A) \triangleq A \cup T(A) \subseteq A,$ which shows that $A$ is $T^{\text {Čech}}$-closed. $(\supseteq)$ Now let $A \subseteq X$ be $T^{\text {Čech}}$-closed. Then $T^{\text {Čech}}(A) \subseteq A,$ and hence $A \cup T(A) \subseteq A,$ which implies $T(A) \subseteq A,$ so $A$ is $T$-closed. ]]

Again, suppose $T$ is a base closure operator. To summarize the results above, we have

$$ T \; \leq \; T^{\text {Čech}} \; \leq \; {\tau}^T{\text {-}}{\text {Cl}} $$

where $T^{\text {Čech}}$ is the Čech modification of $T$ and ${\tau}^T{\text {-}}{\text {Cl}}$ is the identical topological closure operator associated with both $T$ and $T^{\text {Čech}}.$ Lukeš/Malý/Zajíček [36] use the term strong base operator (defined on middle of p. 8 and used frequently thereafter) for a base closure operator $T$ such that $T^{\text {Čech}} = {\tau}^T{\text {-}}{\text {Cl}}$ (the main interest being, of course, when $T < T^{\text {Čech}}),$ which is equivalent (Theorem 1.2 at the bottom of p. 7) to the property that $T^2 \leq T.$ [[ To prevent possible confusion, if $T$ were a Čech closure operator, then as we pointed out at the beginning of this section, the inequality $T^2 \leq T$ would imply that $T$ is a topological closure operator. However, the $T$ that we're considering here is only assumed to be a base closure operator. ]] Moreover, if $T$ is a base closure operator, then $T^{\text {*base}}:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$ defined by $T^{\text {*base}}(A) = {\tau}^T{\text {-}}{\text {Cl}}(T(A))$ is the smallest strong base closure operator greater than $T$ (see top half of p. 9), and thus we can consider $T^{\text {*base}}$ to be the strong base modification of the base closure operator $T.$ Finally, at least for here (see [36] for still more such results), 1.A.8 at the bottom of p. 12 states that for a base closure operator $T,$ the operators $T$ and ${\tau}^T{\text {-}}{\text {Cl}}$ commute $-$ that is, we have $T \circ ({\tau}^T{\text {-}}{\text {Cl}}) = ({\tau}^T{\text {-}}{\text {Cl}}) \circ T.$

Various results about Čech closure operators can be found in many of the references below, but probably the most helpful besides the specific citations above are Čech [9] (especially pp. 237-241, 250-254, 272-275, 555-564, 832-833, 861-863), Kannan [30] (pp. 42-43), Roth [49], and Roth/Carlson [50]. Dolecki/Greco [15] study, among other things, equivalence classes of Čech closure operators that generate the same topological closure operator.

7. Topological Closure Operators $\;$ [satisfy (K1), (K2), (K3), (K4)] ==

Definition: We say that $T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$ is a topological closure operator on $X$ if $T$ is a Čech closure operator on $X$ such that for each $A \in {\mathcal P}(X)$ we have $T(T(A)) = T(A).$

Recall that if $T$ is a Čech closure operator, then $T \leq T^{2}.$ Since the composition of two Čech closure operators is a Čech closure operator, it follows that if $T$ is a Čech closure operator, then we have

$$ T \; \leq \; T^2 \; \leq \; T^3 \; \leq \; T^4 \; \leq \; \cdots, $$

where $\;T^2 = T \circ T,\;$ $\;T^3 = T \circ T \circ T,\;$ etc. are the finite composition iterates of $T.$ This is exactly the same situation we had earlier when we were iterating a monotone closure operator. In this case, each Čech closure operator can be transfinitely iterated to obtain a topological closure operator.

Example (Baire function hierarchy): Let $X = {\mathbb R}^{\mathbb R}$ be the set of functions from $\mathbb R$ to ${\mathbb R}.$ Given ${\mathcal A} \in {\mathcal P}(X)\;$ (thus, $\mathcal A$ is a set of functions from $\mathbb R$ to ${\mathbb R}),\;$ let $T(\mathcal A) = \{f \in X: \; f \; {\text {is a pointwise limit of a sequence of functions in}} \; \mathcal A \}.$ Then $T$ is a Čech closure operator on $X.$ Moreover, if $\mathcal C$ is the set of continuous functions, then $T(\mathcal C)$ is the set of Baire $1$ functions, $T^2(\mathcal C)$ is the set of Baire $2$ functions, etc. In this case $T$ has to be iterated ${\omega}_1$ many times before we get a topology (the topology of pointwise convergence), and ${\tau}^T{\text {-}}{\text {Cl}}(\mathcal C)$ $-$ the corresponding topological closure operator applied to the set of continuous functions $-$ is equal to the set of Borel measurable functions. See this Mathematics Stack Exchange answer by Andrés E. Caicedo for more details.

Topological Modification of a Čech Closure Operator: Let $T$ be a Čech closure operator on $X.$ For each ordinal $\alpha \geq 1$ we define $T^{\alpha}$ by transfinite induction:

$$\begin{array}{l} T^1 & = & T \\ T^{\alpha + 1} & = & T \circ T^{\alpha} \\ T^{\lambda} & = & \sup \,\{T^{\gamma}: \; \gamma < \lambda \}, \;\; {\text{if}} \; \lambda \; {\text {is a nonzero limit ordinal}} \end{array}$$

The following results hold:

(a) $\;\alpha \leq \beta \;$ implies $\;T^{\alpha} \leq T^{\beta}$

(b) $\;$For each ordinal $\alpha \geq 1,$ we have $T^{\alpha}$ is a Čech closure operator on $X.$

(c) $\;$For each ordinal $\alpha \geq 1,$ we have $\;{\tau}^{T^{\alpha}}{\text {-}}{\text {Cl}} = {\tau}^T{\text {-}}{\text {Cl}}.\;$ That is, all the iterates of $T$ have the same corresponding topological closure operator.

(d) $\;$There exists an ordinal $\lambda \leq |X|^{+}$ such that $T^{\alpha} = T^{\lambda}$ for all $\alpha \geq \lambda.$

(e) $\;$Let $|T|$ be the least of the ordinals $\lambda$ in (d) above. If $\;1 \leq \alpha < \beta \leq |T|,\;$ then $\;T^{\alpha} < T^{\beta}.$

(f) $\;$If $\alpha \geq |T|,\;$ then $\;T^{\alpha} = {\tau}^T{\text {-}}{\text {Cl}}.$

(g) $\;$Let $(X,\tau)$ be a topological space and let ${\tau}{\text {-}}{\text {Cl}}$ be the topological closure operator for $(X,\tau).$ Then ${\tau}{\text {-}}{\text {Cl}} = \sup {\mathcal C},$ where $\mathcal C$ is the collection of all Čech closure operators $T$ such that ${\tau}^T{\text {-}}{\text {Cl}} = {\tau}{\text {-}}{\text {Cl}}.$

Regarding (g), note that each of the iterates $T^{\alpha}$ belongs to ${\mathcal C},$ but since there might be other Čech closure operators besides these having ${\tau}{\text {-}}{\text {Cl}}$ for their corresponding topological closure operator, (g) is not automatic from (c) alone. In fact, this possibility can actually arise, and examples can be found in Dolecki/Greco [15].

Various aspects of the above results can be found in Čech [9] (pp. 274-275, 836), Hammer [21], Hausdorff [25], Kannan [30] (pp. 42-43), Kent/Richardson [31], Lukeš/Malý/Zajíček [36] (Exercise 1.A.10 on p. 13), and Novák [45].

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