Limits and Continuous Functions

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

Last updates: 7 April 2011

Filters and limits

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

(a)   if $E\subseteq X$ such that there exists $N\in ℱ$ with $N\subseteq E$ then $E\in ℱ$,

(b)   if $N_1,N_2,\dots ,N_l\in ℱ$ then $N_1\cap N_2\cap \cdots \cap N_l\in ℱ$, and

(c)   $\varnothing \in ℱ$.

Let $Y$ be a topological space. Let $y\in Y$. The neighbourhood filter of $y$ is

$𝒩\left(y\right)=\left\{\text{neighborhoods of}y\right\}$.

A neighbourhood of $y$ is a subset $N\subseteq Y$ such that there exists an open set $U$ with $y\in U\subseteq N$.

Let $Y$ be a topological space and let $𝒢$ be a filter on $Y$.
A limit point of $𝒢$ is a point $y\in Y$ such that $𝒩\left(y\right)\supseteq 𝒢.$
A cluster point of $𝒢$ is a point $y\in Y$ such that if $N\in 𝒢$ then $y\in \bar{N}$, where $\bar{N}$ is the closure of $N$.

Let $X$ be a set with a filter $ℱ$ and let $Y$ be a topogical space. Let $f:X\to Y$ be a function.
A limit point of $f$ is a limit point of the coarsest filter containing $f\left(ℱ\right)$.
A cluster point of $f$ is a cluster point of the coarsest filter containing $f\left(ℱ\right)$.

Write    $y=\underset{ℱ}{\lim }f\left(x\right)$,    if $y$ is a limit point of $f$.

Limits and continuity

Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a function. Let $a\in X$.
A limit of $f$ as $x$ approaches $a$ is a limit point of $f$ with respect to the neighbourhood filter $𝒩\left(a\right)$.

Write     $y=\underset{x\to a}{\lim }f\left(x\right)$,    if $y$ is a limit of $f$ as $x$ approaches $a$.

The function $f:X\to Y$ is continuous at $x=a$ if it satisfies

1. if $N$ is a neighbourhood of $f\left(a\right)$ in $Y$ then $f^{-1}\left(N\right)$ 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 $x=a$ if and only if $$\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right).$$

Sequences

Let $Y$ be a topological space. A sequence $\left(y_1,y_2,y_3,\dots \right)$ in $Y$ is a function $$\begin{array}{rcl}{\mathbb{Z}}_{>0}& ⟶& Y\\ n& ⟼& y_n.\end{array}$$ The Fréchet filter on ${\mathbb{Z}}_{>0}$ is $ℱ=\left\{\text{complements of finite subsets of}{\mathbb{Z}}_{>0}\right\}$.
A limit of the sequence $\left(y_1,y_2,y_3,\dots \right)$ is a limit point of the sequence with respect to the Fréchet filter on ${\mathbb{Z}}_{>0}$.
A cluster point of the sequence $\left(y_1,y_2,y_3,\dots \right)$ is a cluster point with respect to the Fréchet filter on ${\mathbb{Z}}_{>0}$.

Write     $y=\underset{n\to \infty }{\lim }{y}_n$,    if $y$ is a limit point of the sequence $\left(y_1,y_2,y_3,\dots \right)$.

Let $Y$ be a topological space and let $\left(y_1,y_2,y_3,\dots \right)$ be a sequence in $Y$. Let $y\in Y$.

1. $y$ is a limit point of $\left(y_1,y_2,y_3,\dots \right)$ if and only if
   1. if $N_y$ is a neighbourhood of $y$ then there exists $n_0\in {\mathbb{Z}}_{>0}$ such that if $n\in {\mathbb{Z}}_{>0}$ and $n>n_0$ then $y_n\in N_y$.
2. $y$ is a cluster point of $\left(y_1,y_2,y_3,\dots \right)$ if and only if
   1. if $N_y$ is a neighbourhood of $y$ and $n_0\in {\mathbb{Z}}_{>0}$ then there exists $n\in {\mathbb{Z}}_{>0}$ such that $n>n_0$ and $y_n\in N_y$.

Example: The sequence $$y_n=\{\begin{array}{ll}1-\frac{1}{n},& \text{if}n\text{is even,}\\ 0,& \text{if}n\text{is odd,}\end{array}\text{in}ℝ$$ has no limit point, but has cluster points at 1 and 0.

Generating filters

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 ${ℱ}_1\supseteq {ℱ}_2$.

If $ℬ$ is a collection of subsets of $X$ that satisfies

(a)   if $B_1,B_2\in ℬ$ then there exists $B\in ℬ$ such that $B\subseteq B_1\cap B_2$, and

(b)   $ℬ\ne \varnothing$ and $\varnothing \notin ℬ$,

then

$ℱ=\left\{\text{<em>subsets of</em>}X\text{<em>that contain a set in</em>}ℬ\right\}$

is the coarsest filter containing $ℬ$.

Let $X$ be a set and let $𝒮$ be a collection of subsets of $X$. If

$ℬ=\left\{\text{finite intersections of elements of}𝒮\right\}$     and    $ℱ=\left\{\text{subsets of}X\text{containing a set in}ℬ\right\}$

then $ℬ$ satisfies (a) and (b) and $ℱ$ is the coarsest filter on $X$ containing $𝒮$.

Notes and References

This thorough and easy definition of limits, not depending on a metric space, follows [Bou].

References

[GLS] C. Geiss, B. Leclerc and J. Schröer, Semicanonical bases and preprojective algebras, Ann. Sc. École Norm. Sup. 38 (2005), 193-253. (2003), 567-588, arXiv:math/0402448, MR2144987.

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