What is $\mathfrak a$? I'm currently reading Mendelson's Introduction to topology and have came across this theorem:
Theorem 3.8: 
Let a neighborhood in a topological space be defined by Definition 2.2 and an open set in a neighborhood space be defined by Definition 3.5. Then the neighborhoods of a topological space $( X,\mathfrak T)$ give rise to a neighborhood space $ (X,\mathfrak N) = \mathfrak a(X,\mathfrak T)$ and the open sets of a neighborhood space $ (Y,\mathfrak N')$ give rise to a topological space $ (Y,\mathfrak T')=\mathfrak a' (Y,\mathfrak N')$. Furthermore, for each topological space $(X,\mathfrak T)$,
$$(X,\mathfrak T)=\mathfrak a'(\mathfrak a(X,\mathfrak T))$$
and for each neighborhood space $(X,\mathfrak N)$, 
$$ (X,\mathfrak N)=\mathfrak a(\mathfrak a'(X,\mathfrak N)),$$
thus establishing a one-one correspondence between the collection of all topological spaces and the collection of all neighbourhood spaces.
Definition 2.2:
Given a topological space $(X,\mathfrak T)$, a subset $\mathrm N$ of $X$ is called a neighborhood of a point $a \in X$ if $N$ contains an open set that contains $a$.
Definition 3.5:
In a neighborhood space, a subset $O$ is said to be open if it's a neighborhood of each of it's points.
So, my question is:
What are $\mathfrak a$ and $\mathfrak a'$?
 A: The mappings $\mathfrak{a}$ and $\mathfrak{a}'$ are implicitly defined in the statement of the theorem.
$\mathfrak{a}$ maps a topological space $(X,\mathfrak{T})$ to the neighbourhood space $(X,\mathfrak{N})$, where $\mathfrak{N}$ is the family of neighbourhoods of all $x\in X$ in the topology $\mathfrak{T}$, and $\mathfrak{a}'$ is the map in the other direction, that constructs a topological space $(X,\mathfrak{T})$ from a neighbourhood space $(X,\mathfrak{N})$ by letting $\mathfrak{T}$ be the family of sets that are neighbourhoods of all their elements.
The theorem asserts that


*

*the family of neighbourhoods in a topological space satisfy the axioms for a neighbourhood space,

*the family of sets in a neighbourhood space that are neighbourhoods of all their elements satisfy the axioms of a topology,

*going back and forth with these constructions takes you back where you started (i.e. $\mathfrak{a}'$ is the inverse of $\mathfrak{a}$).


Put in another way, neighbourhood spaces and topological spaces are equivalent, and you can choose whichever viewpoint happens to be more convenient.
