variations of Kuratowski closure complement theorem I have been reading about the Kuratowski closure-complement theorem from the paper "THE KURATOWSKI CLOSURE-COMPLEMENT THEOREM  by B.J. Gardner and M. Jackson'. It states that: If $(X,\tau)$ is a topological space and $A \subseteq X$ then at most 14 sets can be obtained from $A$ by taking closures and complements.
Also a direct consequence of this theorem is given which says: When we apply closure and interior on $A \subseteq X$ we get at most 7 sets.This is due to the simple relation $i=kck$ where $i,k,c$ denotes interior,complement,closure operators respectively. Also a figure is given on page 7 of above mentioned paper depicting the interrelationship between the different sets generated.
Presently I was interested in knowing some variations of this theorem. I went through the paper by "VARIATIONS ON KURATOWSKI’S 14-SET THEOREM by DAVID SHERMAN" along with the above mentioned paper.I came across the following result:
If $(X,\tau)$ is a topological space and $A \subseteq X$ then at most 35 sets can be obtained from $A$ by taking closures, interiors, unions, intersections.
Also a diagrammatic representation was given in the paper by Gardener and Jackson on page 36 which depicts the interrelationship between the various sets generated by applying closure, intersection, unions, interior. It shows the containments between the different sets while moving from bottom to top.I have been completely stuck at this figure. I cannot proceed to recognize the various points which correspond to the various operators that are obtained by composing $c,i,\wedge,\vee$ where $\wedge,\vee$ denotes the meet and join operators used to represent the intersection and union of sets respectively. If anyone could please help me to proceed with the figure? 
 A: Sherman's counting scheme is highlighted in Gardner and Jackson's figure below.
First let's review.  Sherman's paper gives the following Hasse diagram for closure $k$, interior $i$ and intersection $\wedge$ (meet), where $I$ is the identity operation on sets:

Then, adding unions $\vee$ (joins), he says:
“Let us add in the two elements $ki\vee ik$ and $(I\wedge ki)\vee(I\wedge ik)$, and partition our poset into four classes:
(1)  $i,k$;
(2)  $I$;
(3)  $iki$, $ki\wedge ik$, $ik$, $ki$, $ki\vee ik$, $kik$;
(4)  $I\wedge iki$, $I\wedge ki\wedge ik$, $I\wedge ik$, $I\wedge ki$, $(I\wedge ki)\vee(I\wedge ik)$, $I\wedge kik$.
It may help to notice that the third class is the right-hand five of [the diagram above] plus $ki\vee ik$, while the fourth class [colored red below] is the middle five plus $(I\wedge ki)\vee(I\wedge ik)$.
Each class above is already a sublattice of [the diagram below], so an irredundant join $x_1\vee x_2\vee\cdots\vee x_n$ can contain at most one $x_j$ from each class. Elements in the first class occur in no irredundant joins. The identity $I$ cannot be involved in an irredundant join except with elements of the third class, which produces six more elements  [colored blue below]. It is left to consider joins of the third and fourth classes. Using the distributive law, this turns up 14 more elements [colored green below].”
The nine elements in classes (1), (2) and (3) are colored black. Note that Gardner and Jackson denote closure by $b$ and the identity operation by $\sf id$.

To prove the existence of a set that generates 35 distinct sets, Sherman gives the following example:
$$\bigg[\bigg\{{1\over n}:n\in\mathbb{N}\bigg\}\bigg]\cup\bigg[[2,4]-\bigg\{3+{1\over n}:n\in\mathbb{N}\bigg\}\bigg]\cup\bigg[(5,7]\cap\bigg(\mathbb{Q}\cup\bigcup_{n=1}^\infty\left(6+{1\over2n\pi},6+{1\over(2n-1)\pi}\right]\bigg)\bigg].$$
Alternatively, Gardner and Jackson construct a finite space with 14 elements that contains a set that generates 35 distinct sets under the four operations. Then they prove that no smaller space can contain such a set.
Fun fact: 14 is also the smallest possible cardinality for a space to contain a set that distinguishes all 86 operations in the meet-semilattice generated by the Kuratowski monoid.*  There exist spaces of cardinality 14 that contain such sets.
*In other words, a seed set that first generates 14 sets under closure and complement, then goes on to also generate the maximum possible number (which turns out to be 72) of new distinct sets obtainable as intersections of the 14 sets, cannot exist in a space with fewer than 14 points.
