Algebra formed by operators and Kuratowski's theorem I have been reading the paper "D. Sherman, Variations on Kuratowski's 14-set theorem, Amer. Math. Monthly 117 (2010), no. 2, 113-123" recently. Kuratowski Closure complement theorem  states that:
 Let $ A \subseteq X$ be a subset of a topological space. The number of distinct
sets which can be obtained from E by successively taking closures and complements
(in any order) is at most 14. Moreover, there are subsets of the Euclidean line
for which 14 is attained.
 The present paper discusses its variations by introducing operations like closure, interior, complement, intersection, union where the corresponding operators are denoted by $\{c, i, k , \vee, \wedge \}$.
The main aim of the author is to apply different operations from this set   repeatedly on a subset of a topological space and obtain various bounds. 
At several places the author uses the term  $\textbf{algebra}$ generated by the topological operations. I have gone through the definition of algebra here https://en.wikipedia.org/wiki/Finitely_generated_algebra. But I can't relate it to the algebra used by the author in this context. If anyone can just help me with this concept?
I have one more question in context to the above mentioned paper. While giving the proofs of the variations of Kuratowski Clsoure complement Theorem with the set of operators $\{c,i,\wedge\}$ or $\{c,i,\wedge, \vee\}$, the author insists on an assumption that topological space contains a copy of $\mathbb{R}$. I do not find any particular use of this assumption in any of these proofs. My question is  if we do not take this assumption can't the proofs just follow? May be  I am not getting it. It would be great if anyone could just help me with this? 
 A: It is used in the sense of universal algebra. Quoting wikipedia

In universal algebra, an algebra (or algebraic structure) is a set A
  together with a collection of operations on A.

A: Sherman's use of the word algebra suppresses the underlying set (on which there is a collection of operations) and refers only to the collection of operations on the underlying set.  More specifically, whenever you see the word algebra in Sherman's paper, you can take it to mean "subset of ${\rm End}({\cal P}(X))$" where $X$ is an arbitrary set, ${\cal P}(X)$ is the power set of $X$, and ${\rm End}({\cal P}(X))$ is the set of operations from ${\cal P}(X)$ to itself.
Note that the underlying set here is ${\cal P}(X)$.  Thus with regard to Sherman's paper anyway, Lord_Farin's comments are misleading on two counts:
(1) They allow the underlying set to be any subset of ${\cal P}(X)$, whereas Sherman's paper always assumes the underlying set is ${\cal P}(X)$ itself.
(2) Instead of suppressing the underlying set (as Sherman does), Lord_Farin suppresses the operations and uses the word algebra to refer to the underlying set.
In Sections 1-4 Sherman explicitly means "subset of ${\rm End}({\cal P}(X))$" when he uses the word algebra; in the rest of the paper he essentially means this same thing, as this critically important paragraph in Section 5 confirms:
"The key observation is that any closure algebra can be identified with a Boolean sublattice of some ${\cal P}(X)$, where $X$ is a topological space and $k$ becomes the associated closure operator [10, Theorem 2.4]. Thus Question 1.3 asks about the cardinalities of subalgebras of $\cal F$ in which only some of $\{I,i,k,c,\wedge,\vee\}$ can be used."
