# 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 $\mathfrak a'$?

• $\mathfrak{a}$ is the mapping that maps each topological space to the corresponding neighbourhood space, and $\mathfrak{a}'$ is the canonical mapping in the other direction. As Theorem 3.8 says, these two maps are inverses of each other. Commented Jul 22, 2013 at 16:25
• @DanielFischer, why don't you turn this into an answer? Commented Jul 22, 2013 at 16:33

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