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Neighborhoods#

Definition: Neighborhood Filter

Let \(X\) be a set and let \(x \in X\).

A neighborhood filter of \(x\) is a non-empty collection \(\mathcal{N}\) of subsets of \(X\), called neighborhoods of \(x\), with the following properties:

  • If \(N \in \mathcal{N}\), then \(x \in N\).
  • If \(N \in \mathcal{N}\) and \(N \subseteq M \subseteq X\), then \(M \in \mathcal{N}\).
  • If \(N_1, \dotsc, N_p \in \mathcal{N}\) for some \(p \in \mathbb{N}_0\), then \(N_1 \cap \cdots \cap N_p \in \mathcal{N}\).

A neighborhood of \(x\) is a subset which "surrounds" \(x\). To properly reflect our intuitive notions of what it means to surround something, neighborhoods must have the additional properties outlined in the above definition:

  • Every neighborhood of \(x\) must contain \(x\) itself, since \(x\) is intuitively contained in all of its surroundings.
  • If a neighborhood \(N\) is contained in some larger subset \(M \subseteq X\), then we also consider the entire \(M\) to be a neighborhood of \(x\). In other words, if \(N\) surrounds \(x\) and is contained in some larger \(M\), then \(M\) itself must surround \(x\).
  • Finally, the intersection of finitely many neighborhoods should itself be a neighborhood. After all, if some areas surround \(x\), then the area where they intersect surely also surrounds \(x\).

A neighborhood filter of \(x\) defines the surroundings of a single \(x \in X\), but we want to do this for each point in \(X\).

Definition: Neighborhood System

A neighborhood system on a set \(X\) is a function \(\mathcal{N}: X \to \mathcal{P}(\mathcal{P}(X))\) which to each \(x \in X\) assigns a neighborhood filter \(\mathcal{N}(x)\) of \(x\) such that if \(N \in \mathcal{N}(x)\), then there exists some \(M \in \mathcal{N}(x)\) with \(M \subseteq N\) and \(N \in \mathcal{N}(y)\) for each \(y \in M\).

A neighborhood system is then just a choice for a neighborhood filter for each \(x \in X\). However, this choice needs to be made in a consistent way, since, on its own, the neighborhood filter of each specific \(x\) is unaware and bears no relation to the neighborhoods of any other points in \(X\). The condition thus links the collections of neighborhoods together and ensures that if a local area \(N\) surrounds \(x\), then it is always possible to find a smaller local area \(M\) surrounding \(x\) such that all points \(y\) in this area \(M\) are surrounded by the larger area \(N\).