Say I build some sort of Topology
If $(X,\mathcal{T})$ is a topological space and $p \in X$, a $\textit{neighbourhood}$ of $p$ is a subset $V$ of $X$, in which $p \in U \subseteq V$, $U$ is open.
We say that $V$ is a $\mathcal{T}-\textit{neighbourhood}$ of $x \in X$ or that $V$ is a $\textit{neighborhood}$ of $x$.
The set of all neighbourhoods of $x \in X$, denoted $\mathcal{N}_x$ is called the $\textit{neighbourhood filter}$ of $x$.
An example of Neighborhood Filters on a Topological space. Let $X = \{a,b,c\}$ and let $\mathcal{T}=\{\emptyset,\{a\},\{b\},\{b,c\},\{a,b\},X\}$ Let $\mathcal{N}_a = \{\{a\},\{a,b\},\{a,c\},X\}, \, \mathcal{N}_b = \{\{b\},\{a,b\},\{b,c\},X\}$, and $\mathcal{N}_{c} = \{\{b,c\},X \}$. In this example $\{a,c \}$ is a neighborhood of a but not of c. Thus a set does not have to be a neighborhood of all of its points.
What is the point of having Neighborhood Filters and what does that tell us about the Topological space, i.e. if a set does not have to be a neighborhood of all of its points, what does this mean for the Topological space? I get the feeling that neighborhoods are some sort of limit which will give us the smallest neighborhood of each point, or perhaps if a neighborhood is not that of all of its points then these points are "seperated" somehow but I am not sure (how it works).
Thanks,
Brian
Best Answer
One can specify a topology in more than four different ways. The standard definition specifies the open sets, what we usually call a "topology." The second way is to specify the close sets-this is of course only a trivial difference. The third way is to specify a closure operation on subsets of your space, and the fourth is to specify a neighborhood filter for every point satisfying the natural axiom that every neighborhood of $x$ is a neighborhood of every point of one of its subsets. So in this sense neighborhood filters tell you everything they possibly could about a topological space.
Probably the best way to think about the neighborhood filter of $x$: is that it contains all information regarding convergence to $x$. In the first topological spaces one encounters, convergence is usually of sequences. But this isn't enough to describe the topology in arbitrary spaces, for instance the infinite-dimensional spaces of functional analysis. It becomes important to speak of convergence of nets, or of filters. A filter on $X$ is just a nontrivial subset of the powerset of $X$ closed under finite intersection and superset, and a filter converges to a point $x$ if and only if it contains the neighborhood filter of $x$. In contrast to the case with sequences, this is enough to specify a topology: in fact it's enough to describe how ultrafilters, that is, maximal filters, converge. So in this sense the neighborhood filter encapsulates the viewpoint that topology generalizes the study of convergent sequences.
Just to comment on your thoughts: in a sense the neighborhood filter describes the smallest neighborhood of a point-except that there is no smallest neighborhood! That's true, at least, in many of the most interesting spaces, and is the main reason to worry about a whole filter of neighborhoods-if there were a smallest neighborhood then in any hypothesis requiring something to hold on a sufficiently small neighborhood of $x$ we could just pick the smallest neighborhood. But the smallest neighborhood of a point must be contained in the intersection of all its neighborhoods, and in, say, a Hausdorff space the intersection of all neighborhoods of $x$ is $\{x\}$, which is not a neighborhood of $x$ when $x$ is not isolated. So the filter functions as a virtual smallest neighborhood of $x$: it doesn't converge to a neighborhood of $x$, so we can't think about its limit, but functionally we do just that.