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I encountered the term neighbourhood space in Bert Mendelson's introduction to topology. The only resource I could find referencing a neighbourhood space on the site is Proof that a neighbourhood space is a topological space under certain condition.

The exact definition given in the book is:

"Let $X$ be a set. For each $x \in X$, let there be given a collection $\eta_x$ of subsets of X (called the neighbourhoods of x), satisfying conditions N1 to N5*. This object is called a neighbourhood space."

* here just refers to the basic properties of neighbourhood, in a topological space: https://proofwiki.org/wiki/Basic_Properties_of_Neighborhood_in_Topological_Space#.7F.27.22.60UNIQ-MathJax-1-QINU.60.22.27.7F:_Point_in_Topological_Space_has_Neighborhood

I am not sure what the structure of this object is supposed to be. For a topological space, we may define it as the tuple $(X,\tau)$, where $X$ is an underlying set and $\tau$ is the topology on $X$, which is just a collection of subsets of $X$. However the book represents a neighbourhood space as $(X,\eta)$, but I am not sure what this really is.

For example, how does it take into account the fact that there is a unique $\eta_x$ for each element of $X$? Is $\eta$ really just the set $\{\eta_x:x \in X\}$? How do I think of what this neighbourhood space object is in terms of sets?

Also I couldn't find too much on neighbourhood spaces, and I'm not sure if perhaps this is an older term; if anyone knows of another term that I could search for that would be very helpful.

masiewpao
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2 Answers2

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In this set-up $\eta_x$ is a collection of subsets of $X$, so a collection of elements of $\mathscr P(X)$ (the power set of $X$) and so an element of $\mathscr P(\mathscr P(X))$. Therefore $\eta$ is a map $X\to\mathscr P(\mathscr P(X))$.

Angina Seng
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I talk about neighbourhood spaces and topologies in this post. There we assign to each point $x$ of $X$ a neighbourhood system $\mathcal{N}(x) \subseteq \mathscr{P}(X)$ satisfying certain axioms. So it consists of a function $\mathcal{N}: X \to \mathscr{P}(\mathscr{P}(X))$ (each point gets a separately assigned $\mathcal{N}(x)$ which is a (non-empty) filter of subsets of $X$, all of whose members contain $x$ etc.)

In said post I show how to define a topology on $X$ from this and how to go back from topologies to neighbourhood spaces in a bijective way.

Henno Brandsma
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