Filters, Limit Points and Cluster Points

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 2 November 2009

Filters

Let $X$ be a set. A filter on $X$ is a collection $ℱ$ of subsets of $X$ such that

if $E \subseteq X$ such that there exists $U \in ℱ$ with $E \supseteq U$ then $E \in ℱ$ ,
finite intersections of elements of $ℱ$ are in $ℱ$ ,
$\emptyset \notin ℱ$ .

Let $X$ be a set and let $ℱ_{1}$ and $ℱ_{2}$ be filters on $X$ . The filter $ℱ_{1}$ is finer than $ℱ_{2}$ if $ℱ_{1} \supseteq ℱ_{2}$ .

Let $X$ be a topological space and let $x \in X$ . The neighbourhood filter of $x$ is the collection $ℱ = \{neighbourhoods of x\} .$

The Fréchet filter on $ℤ_{> 0}$ is the collection $ℱ = \{complements of finite sets of ℤ_{> 0}\} .$

Let $ℱ$ be a filter on a set $X$ . A filter base of $ℱ$ is a collection $ℬ$ of subsets of $X$ such that $ℱ = \{subsets of X that contain a set in ℬ\} .$

Let $ℱ$ be a filter on a set $X$ . A subbase of $ℱ$ is a collection $𝒮$ [SCRIPT S NOT DISPLAYING ON MY MACHINE] of subsets of $X$ such that $ℬ = \{finite intersections of elements of 𝒮\}$ is a base of the filter $ℱ$ .

Limit points and cluster points

Let $X$ be a set and let $ℱ$ be a filter on $X$ . A limit point of $ℱ$ is a point $x \in X$ such that the neighbourhood filter of $x$ is finer than $ℱ$ .

Let $X$ be a set and let $ℬ$ be a filter base of a filter $ℱ$ on $X$ . A cluster point of $ℬ$ is a point $x \in X$ such that $x$ is in the closure of each set in $ℬ$ .

Let $X$ be a set with a filter $ℱ$ and let $Y$ be a topological space. Let $f : X \to Y$ be a function.

A limit point of $f : X \to Y$ is a limit point of the filter base $f (ℱ)$ . Write

$y = lim_{ℱ} f (x)$ if $y$ is a limit point of $f$ .

A cluster point of $f : X \to Y$ is a cluster point of the filter base $f (ℱ)$ .

Let $X$ be a set. A sequence $(x_{1} x_{2} x_{3} \dots)$ of points in $X$ is a function $\begin{array}{rcl} ℤ_{> 0} & ⟶ & X \\ n & ⟼ & x_{n} \end{array}$ .

Let $X$ be a set and let $(x_{1} x_{2} x_{3} \dots)$ be a sequence in $X$ . A limit of the sequence $(x_{1} x_{2} \dots)$ is a limit point of the sequence with respect to the Fréchet filter on $ℤ_{0}$ . Write $y = lim_{n \to \infty} f (x_{n})$ [I ASSUME THIS WAS SUPPOSED TO BE x_n NOT x] if $y$ is a limit of the sequence $(x_{1} x_{2} \dots)$ .

Let $X$ be a set and let $(x_{1} x_{2} \dots)$ be a sequence in $X$ . A cluster point of the sequence $(x_{1} x_{2} \dots)$ is a cluster point of the sequence with respect to the Fréchet filter on $ℤ_{> 0}$ .

Let $X$ and $Y$ be topological spaces. Let $a \in X$ . A limit of $f (x)$ as $x$ approaches $a$ is a limit point of $f$ with respect to the neighbourhood filter of $a$ . Write $y = lim_{x \to a} f (x),$ if $y$ is a limit of $f (x)$ as $x$ approaches $a$ .

Let $X$ and $Y$ be topological spaces and let $a \in X$ . Let $f : X \to Y$ be a function. The function $f$ is continuous at $a$ if it satisfies the condition,

if $N$ is a neighbourhood of $f (x)$ in $Y$ , then $f^{- 1}$ is a neighbourhood of $a$ in $X$ .

Let $X$ and $Y$ be topological spaces and let $a \in X$ . A function $f : X \to Y$ is continuous at $a$ if and only if ${lim}_{x \to a} f (x) = f (a)$ .

Let $X$ be a topological space and let $(x_{1} x_{2} \dots)$ be a sequence in $X$ . Then

$y$ is a limit point of $(x_{1} x_{2} \dots)$ if and only if, if $N_{y}$ is a neighbourhood of $y$ then there exists $n_{0} \in ℤ_{> 0}$ such that $x_{n} \in N_{x}$ for all $n \in ℤ, n \geq n_{0}$ .
$y$ is a cluster point of $(x_{1} x_{2} \dots)$ if and only if, if $N_{y}$ is a neighbourhood of $y$ and $n_{0} \in ℤ_{\geq 0}$ then there exists $n \in ℤ_{> 0}$ with $n \geq n_{0}$ such that $x_{n} \in N_{y}$ .

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)