Metric and Hilbert Spaces

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

Last updated: 4 November 2014

Lecture 46: Convergence

§1 Limit points and cluster points.

§1.1 Filters, nets and sequences.

A directed set is a set $P$ with a relation $\le$ such that

(a)	If $i\in P$ then $i\le i,$
(b)	If $i,j,k\in P$ and $i\le j$ and $j\le k$ then $i\le k,$
(c)	If $i,j\in P$ then there exists $k\in P$ such that $i\le k$ and $j\le k\text{.}$

The favourite example of a directed set if ${\mathbb{Z}}_{>0}$ with $i\le j$ if there exists $n\in {\mathbb{Z}}_{\ge 0}$ with $i+n=j\text{.}$

Let $X$ be a set.

	A filter on $X$ is a collection $ℱ$ of subsets of $X$ such that (a) (upper ideal) if $N\in ℱ$ and $E$ is a subset of $X$ with $N\subseteq E$ then $E\in ℱ,$ (b) (closed under finite intersection) If $\ell \in {\mathbb{Z}}_{>0}$ and $N_1,N_2,\dots ,N_{\ell }$ in $ℱ$ then $$N_1\cap N_2\cap \cdots \cap N_{\ell }\in ℱ,$$ (c) $\varnothing \notin ℱ\text{.}$
$•$	An ultrafilter is a maximal filter $ℱ$ on $X$ (with respect to inclusion).
$•$	A net in $X$ is a function $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& P& ⟶& X\\ & n& ⟼& x_n\end{array}$$ where $P$ is a directed set.
$•$	A sequence in $X$ is a function $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& {\mathbb{Z}}_{>0}& ⟶& X\\ & n& ⟼& x_n\end{array}\text{.}$$

We often write $\stackrel{\rightharpoonup }{x}=(x_1,x_2,\dots )$ for a sequence in $X\text{.}$

Let $P$ be a directed set. The tail filter is the filter on $P$ given by $\left\{P_{\ge N}\hspace{0.17em}\right|\hspace{0.17em}N\in P\},$ where $P_{\ge N}=\{j\in P\hspace{0.17em}|\hspace{0.17em}j\ge N\}\text{.}$

Let $Y$ be a topological space and let $y\in Y\text{.}$ The neighbourhood filter of $y$ is the filter on $Y$ generated by the open sets containing $y\text{.}$

Let $Y$ be a topological space and let $y\in Y\text{.}$

$•$	A neighbourhood of $y$ is a subset $N\subseteq Y$ such that there exists an open set $U\subseteq Y$ with $y\in U\subseteq N\text{.}$
$•$	The neighbourhood filter of $y$ is $$𝒩\left(y\right)=\left\{\text{neighbourhoods of}\hspace{0.17em}y\right\}\text{.}$$

Let $Y$ be a topological space and let $𝒢$ be a filter on $Y\text{.}$

$•$	A limit point of $𝒢$ is a point $y\in Y$ such that $𝒢\supseteq 𝒩\left(y\right)\text{.}$
$•$	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 $Y$ be a topological space. Let $X$ be a set and $f:X\to Y$ be a function. Let $ℱ$ be a filter on $X\text{.}$

$•$	A limit point of $f$ with respect to $ℱ$ is a limit point of the filter on $Y$ generated by $\left\{f\left(N\right)\hspace{0.17em}\right|\hspace{0.17em}N\in ℱ\}\text{.}$
$•$	A cluster point of $f$ with respect to $ℱ$ is a cluster point of the filter on $Y$ generated by $\left\{f\left(N\right)\hspace{0.17em}\right|\hspace{0.17em}N\in ℱ\}\text{.}$

Write $$y=\underset{ℱ}{\text{lim}}f$$ if $y$ is a limit point of $f$ with respect to $ℱ\text{.}$

Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a function. Let $a\in X\text{.}$ Write $$y=\underset{x\to a}{\text{lim}}f\left(x\right)$$ if $y$ is a limit point of $f$ with respect to the neighbourhood filter of $a\text{.}$

Let $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& P& ⟶& X\\ & n& ⟼& x_n\end{array}$$ be a net in $X\text{.}$ Write $$y=\underset{n\to \infty }{\text{lim}}{x}_n$$ if $y$ is a limit point of $\stackrel{\rightharpoonup }{x}$ with respect to the tail filter on $P\text{.}$

Let $\stackrel{\rightharpoonup }{x}=(x_1,x_2,\dots )$ be a sequence in $X\text{.}$ Write $$y=\underset{n\to \infty }{\text{lim}}{x}_n,$$ if $y$ is a limit point of $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& {\mathbb{Z}}_{>0}& ⟶& X\\ & n& ⟼& x_n\end{array}$$ with respect to the tail filter on ${\mathbb{Z}}_{>0}\text{.}$

A cluster point of a sequence $\stackrel{\rightharpoonup }{x}=(x_1,x_2,\dots )$ in $X$ is a cluster point of $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& {\mathbb{Z}}_{>0}& ⟶& X\\ & n& ⟼& x_n\end{array}$$ with respect to the tail filter on ${\mathbb{Z}}_{>0}\text{.}$

Let $X$ be a topological space.

(a)

Let $A\subseteq X\text{.}$ Then $$\begin{array}{rcl}{\displaystyle \bar{A}}& =& {\displaystyle \{z\in X\hspace{0.17em}|\hspace{0.17em}\text{there exists a filter}\hspace{0.17em}ℱ\hspace{0.17em}\text{on}\hspace{0.17em}X\hspace{0.17em}\text{with}\hspace{0.17em}A\in ℱ\hspace{0.17em}\text{and}\hspace{0.17em}\underset{ℱ}{\text{lim}}=z\}}\\ & =& {\displaystyle \{z\in X\hspace{0.17em}|\hspace{0.17em}\text{there exists a net}\hspace{0.17em}\stackrel{\rightharpoonup }{a}:P\to A\hspace{0.17em}\text{with}\hspace{0.17em}\underset{n\to \infty }{\text{lim}}{a}_n=z\}\text{.}}\end{array}$$

(b)

Let $Y$ be a topological space and let $f:X\to Y$ be a function. The following are equivalent. (1) $f:X\to Y$ is continuous. (2) If $ℱ$ is a filter on $X$ and $\underset{ℱ}{\text{lim}}$ exists then $$f\left(\underset{ℱ}{\text{lim}}\right)=\underset{ℱ}{\text{lim}}f\text{.}$$ (3) If $a\in X$ then $\underset{x\to a}{\text{lim}}f\left(x\right)=f\left(a\right)\text{.}$ (4) If $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& P& ⟶& X\\ & n& ⟼& x_n\end{array}$$ is a net in $X$ and $\underset{n\to \infty }{\text{lim}}{x}_n$ exists then $$\underset{n\to \infty }{\text{lim}}f\left(x_n\right)=f\left(\underset{n\to \infty }{\text{lim}}{x}_n\right)\text{.}$$ Notes and References: See [Clark, Cor. 5.9 and Prop. 5.14].

A topological space $X$ is first countable if $X$ satisfies: if $a\in X$ then there exists a countable collection of neighbourhoods of $x$ which generates $𝒩\left(a\right)\text{.}$

Let $X$ be a first countable topological space.

(a)	Let $A\subseteq X\text{.}$ Then $$\bar{A}=\{z\in X\hspace{0.17em}|\hspace{0.17em}\text{there exists a sequence}\hspace{0.17em}(a_1,a_2,\dots )\hspace{0.17em}\text{in}\hspace{0.17em}A\hspace{0.17em}\text{with}\hspace{0.17em}\underset{n\to \infty }{\text{lim}}{\alpha }_n=z\}\text{.}$$
(b)	Let $Y$ be a topological space and let $f:X\to Y$ be a function. The following are equivalent (1) $f$ is continuous. (2) If $\stackrel{\rightharpoonup }{x}=(x_1,x_2,\dots )$ is a sequence in $X$ and $\underset{n\to \infty }{\text{lim}}{x}_n$ exists then $$\underset{n\to \infty }{\text{lim}}f\left(x_n\right)=f\left(\underset{n\to \infty }{\text{lim}}{x}_n\right)\text{.}$$

Let $X$ be a topological space and let $(x_1,x_2,\dots )$ be a sequence in $X\text{.}$

(a)	If $(x_{n_1},x_{n_2},\dots )$ is a subsequence of $(x_1,x_2,\dots )$ and $y=\underset{x\to \infty }{\text{lim}}{x}_{n_k}$ exists then $y$ is a cluster point of $(x_1,x_2,\dots )\text{.}$
(b)	If $X$ is first countable and $y$ is a cluster point of $(x_1,x_2,\dots )$ then there exists a subsequence $(x_{n_1},x_{n_2},\dots )$ of $(x_1,x_2,\dots )$ such that $y=\underset{k\to \infty }{\text{lim}}{x}_{n_k}\text{.}$

Let $X$ be an uncountable set and let $$𝒯=\{A\subseteq X\hspace{0.17em}|\hspace{0.17em}{A}^c\hspace{0.17em}\text{is countable}\}\text{.}$$ Show that

(a)	$X$ is a topological space.
(b)	$X$ is not first countable.
(c)	$X$ is not Hausdorff.
(d)	$X$ is not discrete.
(e)	If $(x_1,x_2,\dots )$ is a sequence in $X$ and $y=\underset{n\to \infty }{\text{lim}}{x}_n$ exists then there exists $N\in {\mathbb{Z}}_{>0}$ such that if $n\in {\mathbb{Z}}_{\ge N}$ then $x_n=y$ i.e., $(x_1,x_2,\dots )$ is eventually constant at $y\text{.}$
(f)	If $A\subseteq X$ and $A$ is uncountable then $\bar{A}=X\text{.}$
(g)	If $A\subseteq X$ then $$\{z\in X\hspace{0.17em}|\hspace{0.17em}\text{there exists a sequence}\hspace{0.17em}(a_1,a_2,\dots )\hspace{0.17em}\text{in}\hspace{0.17em}A\hspace{0.17em}\text{with}\hspace{0.17em}z=\underset{n\to \infty }{\text{lim}}{a}_n\}=A\text{.}$$
(h)	If $A\subseteq X$ is uncountable and $A\ne X$ then $$\bar{A}\ne \{z\in X\hspace{0.17em}|\hspace{0.17em}\text{there exists a sequence}\hspace{0.17em}(a_1,a_2,\dots )\hspace{0.17em}\text{in}\hspace{0.17em}A\hspace{0.17em}\text{with}\hspace{0.17em}z=\underset{n\to \infty }{\text{lim}}{a}_n\}\text{.}$$

Notes and References: [Clark Example 2.1.5]

Notes and References

These are a typed copy of Lecture 46 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on October 19, 2014.

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