Proposition 1

A set is a neighbourhood of each of its points if and only if it is open.

The everyday sense of the word "neighbourhood" is such that many of the properties which involve the mathematical idea of neighbourhood appear as the mathematical expression of intuitive properties; the choice of this term thus has the advantage of making the language more expressive. For this purpose it is also permissible to use the expressions "sufficiently near" and "as near as we please" in some statements. For example, Proposition 1 can be stated in the following form: a set $A$ is open if and only if, for each $x \in A$, all the points sufficiently near $x$ belong to $A$. More generally, we shall say that a property holds for all points sufficiently near a point $x$, if it holds at all points of some neighbourhood of $x$.

Let us denote by $\mathfrak{B}(x)$ the set of all neighbourhoods of $x$. The sets $\mathfrak{B}(x)$ have the following properties:

(V_I) Every subset of $X$ which contains a set belonging to $\mathfrak{B}(x)$ itself belongs to $\mathfrak{B}(x)$.

(V_{II}) Every finite intersection of sets of $\mathfrak{B}(x)$ belongs to $\mathfrak{B}(x)$.

(V_{III}) The element $x$ is in every set of $\mathfrak{B}(x)$.

Indeed, these three properties are immediate consequences of Definition 4 and axiom (O_{II}).

(V_{IV}) If $V$ belongs to $\mathfrak{B}(x)$, then there is a set $W$ belonging to $\mathfrak{B}(x)$ such that, for each $y \in W$, $V$ belongs to $\mathfrak{B}(y)$.

By Proposition 1, we may take $W$ to be any open set which contains $x$ and is contained in $V$.

This property may be expressed in the form that a neighbourhood of $x$ is also a neighbourhood of all points sufficiently near to $x$.

These four properties of the sets $\mathfrak{B}(x)$ are characteristic. To be precise, we have: