Definitions and Examples

Definition (Filter)

A filter on a set $X$ is a collection $\mathcal{F}$ that is

(i) downward directed: $A, B \in \mathcal{F}$ implies there exists $C \in \mathcal{F}$ such that $C \subseteq A \cap B$,
(ii) nonempty: $\mathcal{F} \neq \varnothing$,
(iii) upward closed: $A \in \mathcal{F}$ and $A \subseteq B$ implies $B \in \mathcal{F}$.

An additional property is often useful:
(iv) proper: there exists $A \subseteq X$ such that $A \notin \mathcal{F}$.

Key observations:

  • Being downward directed and upward closed implies that filters are closed under finite intersections, and being proper is equivalent to the requirement that $\varnothing \notin \mathcal{F}$
  • The power set $2^X$ is a filter but not proper.
  • A set that is only downward directed and nonempty is called a filterbase. Any filterbase generates a filter simply by taking the upward closure of the base.
Example

Trivial filter: The smallest filter $\{X\}$ on any set $X$.

Eventuality filter: For a sequence $\{x_n\}$ in a space $X$:

$$\mathcal{E}_{x_n} = \{A \subseteq X \mid \exists\, N \text{ such that } x_n \in A \text{ for all } n \geq N\}$$

Non-example: Open neighbourhoods of a point $x$ form a filterbase, not a proper filter.

Definition (Convergence)
A filter $\mathcal{F}$ on a topological space $(X,\mathcal{T})$ converges to $x$ if and only if $\mathcal{F}$ refines $\mathcal{T}_x$, that is, if $\mathcal{T}_x \subseteq \mathcal{F}$. When $\mathcal{F}$ converges to $x$ we will write $\mathcal{F} \rightarrow x$.

Remark: Sequences converge to $x$ precisely when their eventuality filter converges to $x$.

Main Theorem

With the concept of filters, we now give an equivalence statement between the categorical limit and the topological limit.

Let $X$ be a topological space and $\mathcal{F}_X$ be the collection of all filters on $X$.

Given $x \in X$ and $F \in \mathcal{F}_X$, let $\mathcal{U}_X(x)$ be the neighborhood filter of $x$ and let

$$ \mathcal{F}_{x,F} = \lbrace G \in \mathcal{F}_X \mid \mathcal{F} \cup \mathcal{U}_X(x) \subseteq G \rbrace $$

be the collection of all the filters containing both $F$ and the neighborhood filter of $x$.

We observed that $F \subseteq F \cup \mathcal{U}_X(x) \subseteq \bigcap_{G\in \mathcal{F}_{x,F}}G$, and the latter two sets are filters.

Also since $\bigcap_{G\in \mathcal{F}_{x,F}}G$ is the smallest filter containing $F \cup \mathcal{U}_X(x)$ and $F \cup \mathcal{U}_X(x)$ is itself a filter, we have that $\bigcap_{G\in \mathcal{F}_{x,F}}G \subseteq F \cup \mathcal{U}_X(x)$ so that $\bigcap_{G\in \mathcal{F}_{x,F}}G = F \cup \mathcal{U}_X(x)$.

View the inclusion of sets “$\subseteq$” as a partial order on $\mathcal{F}_X$ and $\mathcal{F}_{x,F}$. Then $\mathcal{F}_X$ becomes a small category and $\mathcal{F}_{x,F}$ is its subcategory. Let $E: \mathcal{F}_{x,F} \rightarrow \mathcal{F}_X$ be the embedding functor. Now we have a theorem indicating the connection between the categorical limit and the topological limit.

Theorem
$x$ is a topological limit of filter $\mathcal{F}$ if and only if $\mathcal{F}$ is a categorical limit of the embedding functor $E \colon \mathcal{F}_{x,\mathcal{F}} \to \mathcal{F}_X$.
Proof
$x$ is a topological limit of the filter $F$, i.e., $F \to x$
$\Leftrightarrow (\mathcal{T}_x \subseteq \mathcal{U}(x)) \subseteq F \ \text{and} \ F \cup \mathcal{U}(x) \subseteq F$
$\Leftrightarrow F = \bigcap_{G \in \mathcal{F}_{X,F}} g$ (the universal object)
$\Leftrightarrow (F \xrightarrow{p_{G}}E(G))_ {G \in \mathcal{F}_ {x,F}}$ is a limit cone of $E$
(i.e., $F$ is a categorical limit of $E$)□

References