|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-angg "LATEX/2008filterp-slides.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2008filterp-slides.tex && latex 2008filterp-slides.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2008filterp-slides.tex && pdflatex 2008filterp-slides.tex"))
% (eev "cd ~/LATEX/ && Scp 2008filterp-slides.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (find-dvipage "~/LATEX/2008filterp-slides.dvi")
% (find-pspage "~/LATEX/2008filterp-slides.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2008filterp-slides.ps 2008filterp-slides.dvi")
% (find-pspage "~/LATEX/2008filterp-slides.ps")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2008filterp-slides.pdf" (ee-twupfile "LATEX/2008filterp-slides.pdf") 'over)
% (ee-cp "~/LATEX/2008filterp-slides.pdf" (ee-twusfile "LATEX/2008filterp-slides.pdf") 'over)
% «.nsa-main-idea» (to "nsa-main-idea")
% «.nsa-2» (to "nsa-2")
% «.filters-1» (to "filters-1")
% «.proper-bsm-ultra» (to "proper-bsm-ultra")
% «.some-sentences» (to "some-sentences")
% «.cores-and-principal» (to "cores-and-principal")
% «.ultras-are-evil» (to "ultras-are-evil")
% «.diagram» (to "diagram")
% «.big-domains» (to "big-domains")
% «.filters-are-enough» (to "filters-are-enough")
\documentclass[oneside]{book}
\usepackage[latin1]{inputenc}
\usepackage{edrx08} % (find-dn4ex "edrx08.sty")
%L process "edrx08.sty" -- (find-dn4ex "edrx08.sty")
\input edrxheadfoot.tex % (find-dn4ex "edrxheadfoot.tex")
\begin{document}
\input 2008filterp-slides.dnt
%*
% (eedn4-51-bounded)
Index of the slides:
\msk
% To update the list of slides uncomment this line:
\makelos{tmp.los}
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")
\tocline {Non-Standard Analysis} {2}
\tocline {Non-Standard Analysis (2)} {3}
\tocline {Filters} {4}
\tocline {Proper filters, big/small/medium sets, and ultrafilters} {5}
\tocline {Cores and principal ultrafilters} {6}
\tocline {Interpreting some sentences} {7}
\tocline {Ultrafilters are evil} {8}
\tocline {Partial functions with big domains} {9}
\tocline {Diagram} {10}
\tocline {Filters are enough} {11}
% --------------------
% defs
\def\SetI{\Set^\I}
\def\SetN{\Set^\N}
\def\SetIF{\Set^\I/\F}
\def\SetIU{\Set^\I/\U}
\def\SetNN{\Set^\N/\calN}
\def\SetNU{\Set^\N/\U}
\def\simF{\sim_\F}
\def\simN{\sim_\N}
\def\simU{\sim_\U}
\def\ph{\leavevmode\phantom}
\def\Def:{{\bf Def:}}
\def\ind{\ph{\Def:} }
\def\Opens{\mathcal{O}}
\def\calN{{\mathcal{N}}}
\def\calM{{\mathcal{M}}}
\def\calR{{\mathcal{R}}}
\def\calV{{\mathcal{V}}}
\def\calX{{\mathcal{X}}}
\def\calY{{\mathcal{Y}}}
\def\calZ{{\mathcal{Z}}}
\def\Seti#1{\Set^{(-#1,#1)}}
\def\Setf#1{\Set^{(-\frac{1}{#1},\frac{1}{#1})}}
\def\simnat{\overset{î}{\sim}}
\def\iff{\Leftrightarrow}
\def\V{{\mathcal{V}}}
\def\X{{\mathcal{X}}}
\def\Y{{\mathcal{Y}}}
\def\mathbblow{\mathbbold}
\def\XX{(X,\X)}
\def\YY{(Y,\Y)}
\def\XXz{(X,\X_{x_0})}
\def\YYz{(Y,\Y_{y_0})}
\def\IF{(\I,\F)}
\def\IU{(\I,\U)}
\def\NN{(\N,\calN)}
\def\RRz{(\R,\calR_0)}
\newpage
% --------------------
% «nsa-main-idea» (to ".nsa-main-idea")
% (s "Non-Standard Analysis" "nsa-main-idea")
\myslide {Non-Standard Analysis} {nsa-main-idea}
The main idea:
$\Set$ is the ``standard universe'',
$\SetN$ is the ``universe of ($\N$-)sequences'',
$\SetNN$ is the ``universe of $\N$-sequences modulo $\sim_\calN$'',
$\SetNU$ is the ``universe of $\N$-sequences modulo $\sim_\U$'',
where $\sim_\calN$ is the equivalence relation induced by the filter $\calN$,
and $\sim_\U$ is the equivalence relation induced by the ultrafilter $\U$,
where $\sim_\U$ has bigger classes than $\sim_\N$.
%D diagram unnamed-arrows-N
%D 2Dx 100 +30 +30
%D 2D 100 \Set ---> \Set^\N --> \SetNN
%D 2D \ :
%D 2D \ :
%D 2D v v
%D 2D +30 \SetNU
%D 2D
%D (( \Set \Set^\N \SetNN
%D \SetNU
%D @ 0 @ 1 -> @ 1 @ 2 -> @ 1 @ 3 -> @ 2 @ 3 .>
%D ))
%D enddiagram
$$\diag{unnamed-arrows-N}$$
$\Set \to \SetN$ takes 4 to $(4,4,4,4,\ldots)$,
$\SetN \to \SetNN$ takes
$(1,\frac12,\frac13,\frac14,\ldots)$ to
$(1,\frac12,\frac13,\frac14,\ldots)/\calN$, and
equivalence classes of sequences tending to zero will
behave as infinitesimals.
\msk
$\SetNU$ is a ``non-standard universe''.
$\Set^\N$ and $\SetNU$ are quite similar ---
they both obey the same first-order formulas (!!!)
(with bounded quantifiers and all constants standard)
and we have ``transfer theorems'' that let us ``transfer truths''
from $\Set$ to $\SetNU$ and back.
And $\SetNU$ has infinitesimals!!!
\newpage
% --------------------
% «nsa-2» (to ".nsa-2")
% (s "Non-Standard Analysis (2)" "nsa-2")
\myslide {Non-Standard Analysis (2)} {nsa-2}
The general case:
$\Set$ is the ``standard universe'',
$\SetI$ is the ``universe of ($\I$-)sequences'',
$\SetIF$ is the ``universe of $\I$-sequences modulo $\sim_\F$'',
$\SetIU$ is the ``universe of $\I$-sequences modulo $\sim_\U$'',
where $\sim_\F$ is the equivalence relation induced by the filter $\F$,
and $\sim_\U$ is the equivalence relation induced by the ultrafilter $\U$,
where $\sim_\U$ has bigger classes than $\sim_\F$.
%D diagram unnamed-arrows
%D 2Dx 100 +30 +30
%D 2D 100 \Set ---> \Set^\I --> \Set^\I/\F
%D 2D \ :
%D 2D \ :
%D 2D v v
%D 2D +30 \Set^\I/\U
%D 2D
%D (( \Set \Set^\I \Set^\I/\F
%D \Set^\I/\U
%D @ 0 @ 1 -> @ 1 @ 2 -> @ 1 @ 3 -> @ 2 @ 3 .>
%D ))
%D enddiagram
$$\diag{unnamed-arrows}$$
$\F$ is a filter on the index set $\I$,
$\U$ is an ultrafilter on $\I$, refining $\F$ (i.e., $\F \subset \U$).
% ``semi-standard universe''
\newpage
% --------------------
% «filters-1» (to ".filters-1")
% (s "Filters" "filters-1")
\myslide {Filters} {filters-1}
{\bf Definition:} $\F \subseteq \Pts(\I)$ is a filter on $\I$ iff:
\ssk
(i) $\I \in \F$,
(ii) $\F$ is closed by binary intersections,
(iii) $\F$ is ``closed by supersets''.
\msk
Our two archetypical filters:
%
$$\begin{array}{l}
\calN \subset \Pts(\N) \\
\calN := \sst{I \subset \N}{\N \bsl I \text{ is finite}} \\
\calR_0 \subset \Pts(\R) \\
\calR_0 := \sst{I \subset \R}{I \text{ contains an open neighborhood of 0}} \\
\end{array}
$$
$\calN$ is the ``filter of cofinites'' (on $\N$),
$\calR_0$ is the ``filter of neighborhoods of 0'' (in $\R$).
\msk
Define the following relation on $\I$-sequences:
$$a \simF b \quad \Bij \quad \sst{i}{a_i = b_i} \in \F$$
\msk
Prop: $\simF$ is an equivalence relation $\funto$ $\F$ is a filter.
\msk
$\begin{array}{lcl}
a \simF a & \funto & \F \ni \sst{i}{a_i = a_i} = \I, \\
a \simF b \simF c & \funto & \F \ni \sst{i}{a_i = c_i} \supseteq
\sst{i}{a_i = b_i} Ì \sst{i}{b_i = c_i}, \\
\end{array}
$
\msk
Look at this example (with $\I := \R$):
$f$ is 0 in $(-2,1)$, 1 elsewhere,
$g$ is 0 everywhere,
$h$ is 0 in $(-1,2)$, $-1$ elsewhere,
$h'$ is 0 in $(-1,2)$, 1 in $(4,5)$, $-1$ elsewhere;
\ssk
$f$ coincides with $h$ exactly on $(-2,1)Ì(-1,2)$,
$f$ coincides with $h'$ on a bigger set --- the above plus $(4,5)$.
\msk
Prop: $\simF$ is an equivalence relation $\funot$ $\F$ is a filter.
\newpage
% --------------------
% «proper-bsm-ultra» (to ".proper-bsm-ultra")
% (s "Proper filters, big/small/medium sets, and ultrafilters" "proper-bsm-ultra")
\myslide {Proper filters, big/small/medium sets, and ultrafilters} {proper-bsm-ultra}
\Def: a filter $\F$ is {\sl proper} when $\emp \notin \F$.
\ind $\F$ improper $\Bij$ $\emp \in \F$ $\Bij$ $\F = \Pts(\I)$ $\Bij$
\ind $\Bij$ all sequences are $\F$-equivalent.
\ind $\calN$ is proper.
\msk
\Def: $I \subset \I$ is {\sl $\F$-big} when $I \in \F$.
\ind $\N+4 = \{4,5,6,7,\ldots\}$ is cofinite, and so $\calN$-big.
\Def: $I \subset \I$ is {\sl $\F$-small} when $I \in \F$.
\ind $\{0,1,2,3\}$ is finite, and so $\calN$-small.
\Def: $I \subset \I$ is {\sl $\F$-medium} when $I$ is neither $\F$-big, nor $\F$-small.
\ind $2\N = \{0,2,4,6,...\}$ is $\calN$-medium.
\msk
A proper filter $\F$ divides $\Pts(\I)$ in $\F$-big, $\F$-medium and $\F$-small sets.
\Def: an {\sl ultrafilter} is a filter $\F$ with no $\F$-medium sets.
\ind We will use $\U$ to denote ultrafilters.
\ind $\calN$ is not an ultrafilter.
\msk
Two proper filters over $\I := \{\aa,\bb,\cc\}$:
The one at the right is an ultrafilter.
%
%D diagram 3cube
%D 2Dx 100 +25 +25 +25 +25 +25 +25 +25 +25
%D 2D 100 111 111a 111b
%D 2D / | \ / | \ / | \
%D 2D +20 011 101 110 011a 101a 110a 011b 101b 110b
%D 2D | X X | | X X | | X X |
%D 2D +20 001 010 100 001a 010a 100a 001b 010b 100b
%D 2D \ | / \ | / \ | /
%D 2D +20 000 000a 000b
%D 2D
%D (( 111 011 - 111 101 - 111 110 -
%D 011 001 - 011 010 - 101 001 - 101 100 - 110 010 - 110 100 -
%D 001 000 - 010 000 - 100 000 -
%D ))
%D (( 000a .tex= S 001a .tex= M 010a .tex= M 011a .tex= B
%D 100a .tex= S 101a .tex= M 110a .tex= M 111a .tex= B
%D 111a 011a - 111a 101a - 111a 110a -
%D 011a 001a - 011a 010a - 101a 001a - 101a 100a - 110a 010a - 110a 100a -
%D 001a 000a - 010a 000a - 100a 000a -
%D ))
%D (( 000b .tex= S 001b .tex= B 010b .tex= S 011b .tex= B
%D 100b .tex= S 101b .tex= B 110b .tex= S 111b .tex= B
%D 111b 011b - 111b 101b - 111b 110b -
%D 011b 001b - 011b 010b - 101b 001b - 101b 100b - 110b 010b - 110b 100b -
%D 001b 000b - 010b 000b - 100b 000b -
%D ))
%D enddiagram
%D
$$\diag{3cube}$$
For $\calA \subset \Pts(\I)$,
\Def: $\upto \calA := \sst{A'}{A \subseteq A' \subseteq \I, \text{for some $A \in \calA$}}$
\ind $\upto \F = \F$.
\Def: $\dnto \calA := \sst{A'}{A' \subseteq A, \text{for some $A \in \calA$}}$
\ind The set of $\F$-small sets is equal to its `$\dnto$'.
\Def: $\interfin \calA := \sst{A_1Ì\ldotsÌA_n}{n\in\N, A_1,\ldots,A_n \in \calA}$
\ind where we define that $A_1Ì\ldotsÌA_n = \I$ when $n=0$.
\msk
{\bf Fact:} for any $\calA \subset \Pts(\I)$,
\ph{\bf Fact:} $\interfin \upto \A = \upto \interfin \A$ is a filter.
\bsk
$\calN = \upto \interfin \{ \N, \N+1, \N+2, \N+3, \ldots \}$
$\calR_0 = \upto \interfin \{ (-1,1), \, (-\frac12,-\frac12), \, (-\frac13,-\frac13), \ldots \}$
\newpage
% --------------------
% «cores-and-principal» (to ".cores-and-principal")
% (s "Cores and principal ultrafilters" "cores-and-principal")
\myslide {Cores and principal ultrafilters} {cores-and-principal}
The {\sl core} of a filter $\F$ is $\bigcap\F$.
$\calN$ has empty core.
$\calR_0$ has core $= \{0\}$, but this can be ``fixed'' ---
by removing $\{0\}$ from each $\calR_0$-big set we get a
filter over $\R\bsl\{0\}$ --- the filter of
``punctured neighborhoods'' of $0 \in \R$, that has
empty core.
\msk
(By the way: $\calN$ is a filter of punctured
neighborhoods of $‚Ý\N^*$ in $\N^*\bsl\{‚\}$.)
\msk
Any ultrafilter refining $\calN$ has empty core.
An ultrafilter with a non-empty core has a single point in its core.
An ultrafilter with a non-empty core is called ``principal''.
Principal ultrafilters are silly: if $\U = \upto\{a\}$
then the equivalence relation $\sim_\U$ pays attention only
to the index $a$, and $\Set \cong \SetIU$.
\msk
$$\diag{unnamed-arrows-N}$$
\msk
When $\U$ is non-principal
every infinite set in $\Set$
gets new (``non-standard'') elements
after the passage to $\SetIU$.
\newpage
% --------------------
% «some-sentences» (to ".some-sentences")
% (s "Interpreting some sentences" "some-sentences")
\myslide {Interpreting some sentences} {some-sentences}
Take $Ï:=(1,2,3,4,\ldots)$ in $\SetNN$.
$Ï$ is bigger than any standard natural:
$Ï>2 \equiv (\False,\False,§,§,\ldots) \sim_\calN (§,§,§,§,\ldots) \equiv §$
\msk
Take $:=(1,\frac12,\frac13,\frac14,\ldots)$ in $\SetNN$.
$$ is smaller than any standard positive real:
$<\frac12 \equiv (\False,\False,§,§,\ldots) \sim_\calN §$.
\msk
$f(a)$ is $(f_1(a_1), f_2(a_2), f_3(a_3), \ldots)$.
\msk
$ýa,bÝ\R. ab=ba$
\msk
$ýxÝ(0,1).x^2Ý(0,x)$
\msk
$ýa,bÝ\R.ab=0 ⊃ (a=0 ∨ b=0)$
\newpage
% --------------------
% «ultras-are-evil» (to ".ultras-are-evil")
% (s "Ultrafilters are evil" "ultras-are-evil")
\myslide {Ultrafilters are evil} {ultras-are-evil}
Take a denumerable family of sets of indices, $\calA = \{A_1, A_2, A_3, \ldots\}$,
for example $\calA := \{\N, 2\N, 3\N, 4\N, \ldots\}$.
Then $\upto \interfin \calA$ is not a non-principal ultrafilter.
Let's see why.
Take $\calA' := \{A_1, A_1ÌA_2, A_1ÌA_2ÌA_3, \ldots\}$;
build $\calA''$ from that by removing the repetitions.
In the non-trivial case, $\calA'' = \{A''_1, A''_2, A''_3, \ldots\}$ is infinite.
Look at
$(\I \bsl A''_1) þ (A''_2 \bsl A''_3) þ (A''_4 \bsl A''_5) þ \ldots$ and
$(A''_1 \bsl A''_2) þ (A''_3 \bsl A''_4) þ (A''_5 \bsl A''_6) þ \ldots$ ---
they are both medium sets.
\msk
Attempts to build non-principal explicitly are bound to fail.
To build non-principal ultrafilters we need a weak form of AC.
Halpern 1964: the ``boolean prime ideal theorem'' is independent from AC.
\newpage
% --------------------
% «big-domains» (to ".big-domains")
% (s "Partial functions with big domains" "big-domains")
\myslide {Partial functions with big domains} {big-domains}
If $(X,\calX)$ and $(Y,\calY)$ are filtered spaces ---
i.e., $\calX$ is a filter over $X$
and $\calY$ is a filter over $Y$ ---
then a partial function $f:X \to Y$ is said
to have ($\calX$-)big domain when its domain is $\calX$-big.
\msk
Shorter name: a ``big partial function'' is a
partial function with a big domain.
Even shorter: $\to$ ``big function''.
\msk
{\bf Filter-continuity}
A partial function $f:X \to Y$ is {\sl (filter-)continuous} when
the inverse image of every $\calY$-big set is $\calX$-big.
(Being ``big'' is weaker than that: just $f^{-1}(Y) Ý \calX$.)
\msk
Two big functions $f,g$ are {\sl equivalent} when
they coincide on a big set.
\msk
Big continuous functions compose.
Moreover: if $f \sim_{\calX} f'$ and $g \sim_{\calY} g'$ are all big and continuous,
then $g¢f \sim_{\calX} g'¢f'$ is big and continuous.
%D diagram filtermapcomp
%D 2Dx 100 +50
%D 2D 100 (X,\calX)
%D 2D
%D 2D +40 (Y,\calY) (Z,\calZ)
%D 2D
%D (( (X,\calX) (Y,\calY) (Z,\calZ)
%D @ 0 @ 1 -> sl_ .plabel= l f
%D @ 0 @ 1 -> sl^ .plabel= r f'
%D @ 1 @ 2 -> sl^ .plabel= a g'
%D @ 1 @ 2 -> sl_ .plabel= b g
%D @ 0 @ 2 .>
%D ))
%D enddiagram
%D
$$\diag{filtermapcomp}$$
\newpage
% --------------------
% «diagram» (to ".diagram")
% (s "Diagram" "diagram")
\myslide {Diagram} {diagram}
\def\aw{\frac aÏ}
%D diagram wo-t0
%D 2Dx 100 +50 +50 +30 +45
%D 2D 100 \o |---> g_3
%D 2D - ||
%D 2D +15 | || \aw |---> log(1+\aw)
%D 2D v || |--> || ||
%D 2D +15 \o,\O |--> g_4 Ï || ||
%D 2D - || - |--> || ||
%D 2D +15 | || | {}\o |---> log(1+\o)
%D 2D | || v - ||
%D 2D +15 | g_5 Ï,\o' | ||
%D 2D v |-> || |--> v ||
%D 2D +15 \o,\o' || {}\o,\o' |-> (1+\o')\o
%D 2D |-> ||
%D 2D +15 g_6
%D 2D
%D (( g_3 .tex= f(b+\o)
%D g_4 .tex= f(b)+f'(b)\o+\O\o^2
%D g_5 .tex= f(b)+f'(b)\o+\o'\o
%D g_6 .tex= f(b)+(f'(b)+\o')\o
%D ))
%D (( \o \o,\O \o,\o'
%D @ 0 @ 1 |-> @ 1 @ 2 |->
%D @ 0 g_3 |-> @ 1 g_4 |-> @ 2 g_5 |-> @ 2 g_6 |->
%D g_3 g_4 = g_4 g_5 = g_5 g_6 =
%D ))
%D (( \aw log(1+\aw) # 0 1
%D Ï {}\o log(1+\o) # 2 3 4
%D Ï,\o' {}\o,\o' (1+\o')\o # 5 6 7
%D @ 2 @ 5 |->
%D @ 0 @ 3 = @ 3 @ 6 |->
%D @ 1 @ 4 = @ 4 @ 7 =
%D @ 2 @ 0 |-> @ 0 @ 1 |->
%D @ 2 @ 3 |-> @ 3 @ 4 |->
%D @ 5 @ 6 |-> @ 6 @ 7 |->
%D ))
%D enddiagram
%D diagram wo-t1
%D 2Dx 100 +30 +45
%D 2D 100
%D 2D
%D 2D +15
%D 2D
%D 2D +15
%D 2D
%D 2D +15
%D 2D
%D 2D +15
%D 2D
%D enddiagram
%D diagram wo-t
%D 2Dx 100 +45 +45
%D 2D 100 h_1 |---> h_5
%D 2D |---> || ||
%D 2D +15 Ï || ||
%D 2D - |---> || ||
%D 2D +15 | h_2 ||
%D 2D | || ||
%D 2D +15 | || ||
%D 2D v || ||
%D 2D +15 Ï,\o' |-> h_3 ||
%D 2D - || ||
%D 2D +15 | || ||
%D 2D | || ||
%D 2D +15 | -> h_4 |---> h_6
%D 2D | / ||
%D 2D +15 | / ||
%D 2D v \ ||
%D 2D +15 \o' |------------> h_7
%D 2D / ||
%D 2D +15 \ ||
%D 2D \ ||
%D 2D +15 \-> \o'' |--> h_8
%D 2D - ||
%D 2D +15 | ||
%D 2D v ||
%D 2D +15 \o''' |--> h_9
%D 2D
%D (( h_1 .tex= \log(1+\aw)^Ï h_5 .tex= (1+\aw)^Ï
%D h_2 .tex= Ï\log(1+\aw)
%D h_3 .tex= Ï((1+\o')\aw)
%D h_4 .tex= (1+\o')a h_6 .tex= e^{(1+\o')a}
%D h_7 .tex= e^{a+\o'a}
%D h_8 .tex= e^{a+\o''}
%D h_9 .tex= e^a+\o'''
%D ))
%D (( Ï Ï,\o' \o' \o'' \o'''
%D @ 0 @ 1 |-> @ 1 @ 2 |-> @ 2 @ 3 |-> @ 3 @ 4 |->
%D ))
%D (( h_1 h_2 = h_2 h_3 = h_3 h_4 =
%D h_5 h_6 = h_6 h_7 = h_7 h_8 = h_8 h_9 =
%D h_1 h_5 |-> .plabel= a \exp
%D h_4 h_6 |-> .plabel= a \exp
%D ))
%D (( Ï h_1 |-> Ï h_2 |->
%D Ï,\o' h_3 |->
%D \o' h_4 |-> \o' h_7 |->
%D \o'' h_8 |->
%D \o''' h_9 |->
%D ))
%D ((
%D
%D ))
%D enddiagram
$$\diag{wo-t0}$$
\msk
$$\diag{wo-t}$$
\newpage
% --------------------
% «filters-are-enough» (to ".filters-are-enough")
% (s "Filters are enough" "filters-are-enough")
\myslide {Filters are enough} {filters-are-enough}
% (find-LATEXfile "2008filterp.tex" "%D diagram wo")
Main theorem
Change of base
% (find-LATEX "2008filterp.tex" "natural-infinitesimals")
Filter-continuity is the same as continuity at the chosen point:
$$(\R,\calR_0) \to (X,\calX_{x_0})$$
\msk
Filter-continuity is the same as infinitesimality:
$$(\I,\F) \to (\R,\calR_0)$$
\msk
(general case: topological spaces)
Definition: the {\sl natural infinitesimal} on a (standard) filtered
space $(X,\X_{x_0})$, that we will denote by $x_1^î \simnat x_0$, is
the identity function $x_1^î = \id: (X,\X_{x_0}) \to (X,\X_{x_0})$;
seen as an infinitesimal, it lives in $\Set^X/\X_{x_0}$. As it
corresponds to the identity map, any other infinitesimal $x_1 \sim
x_0$ --- in the diagram below we take an $x_1$ living in $\Set^\I/\F$
--- factors through $x_1^î$ it in a unique way; this suggests that
there is a kind of ``change of base'' operation between filter-powers.
%D diagram nat-infinitesimal
%D 2Dx 100 +35
%D 2D 100 (\I,\F) ..> (X,\X_{x_0}){}
%D 2D \ |
%D 2D v v
%D 2D +20 (X,\X_{x_0})
%D 2D
%D (( (\I,\F) (X,\X_{x_0}){} (X,\X_{x_0})
%D @ 0 @ 1 .> .plabel= a x_1
%D @ 0 @ 2 -> .plabel= l x_1
%D @ 1 @ 2 -> .plabel= r x_1^î=\id
%D ))
%D enddiagram
%D
$$\diag{nat-infinitesimal}$$
% Our notation for it will be: $x_1 \simnat x_0$.
Now, for any $f: (X,\calX_{x_0}) \to (Y,\calY_{y_0})$ taking $x_0$ to
$y_0$, this holds:
% \smallskip
\begin{quotation}
{\bf Key theorem:}
(i) $f$ is continuous at $x_0$
$\iff$ (ii) for $(\I,\F) := (X,\calX_{x_0}),$ $x^î_1 \simnat x_0$, we have $f(x^î_1) \sim f(x_0)$
$\iff$ (iii) for all $(\I,\F)$ and $x_1 \sim x_0$, we have $f(x_1) \sim f(x_0)$.
% $\iff$ (iv) for all $(\I,\U)$ and $x_1 \sim x_0$, we have $f(x_1) \sim f(x_0)$.
\end{quotation}
%D diagram keyth-diags-1
%D 2Dx 100 +20 +35 +15 +20 +30
%D 2D 100 A0 a0
%D 2D | |
%D 2D x1î | x1î |
%D 2D v f v f
%D 2D +25 A1 -> A2 a1 -> a2
%D 2D
%D 2D +20 B0 b0
%D 2D \ \
%D 2D x1 \ x1 \
%D 2D v f v f
%D 2D +25 B1 -> B2 b1 -> b2
%D 2D
%D (( A0 .tex= (X,\X_{x_0}) A1 .tex= (X,\X_{x_0}) A2 .tex= (Y,\Y_{y_0})
%D B0 .tex= (\I,\F) B1 .tex= (X,\X_{x_0}) B2 .tex= (Y,\Y_{y_0})
%D a0 .tex= x a1 .tex= x a2 .tex= y
%D b0 .tex= i b1 .tex= x b2 .tex= y
%D A0 A1 -> .plabel= l x_1^î A0 A2 -> .plabel= a y_1 A1 A2 -> .plabel= r f
%D B0 B1 -> .plabel= l x_1 B0 B2 -> .plabel= a y_1 B1 B2 -> .plabel= r f
%D a0 a1 |-> .plabel= l x_1^î a0 a2 |-> .plabel= a y_1 a1 a2 |-> .plabel= r f
%D b0 b1 |-> .plabel= l x_1 b0 b2 |-> .plabel= a y_1 b1 b2 |-> .plabel= r f
%D ))
%D enddiagram
%D
$$\diag{keyth-diags-1}$$
% \smallskip
Proof: (i) $\funto$ (ii) and (i) $\funto$ (iii) are obvious from what
we've seen before --- that the composite of continuous maps between
filtered spaces is continuous. For $¬$(i) $\funto$ $¬$(ii), as $f$ is
not continuous at $x_0$, we can choose a $Y' \in \Y_{y_0}$ such that
$f^{-1}(Y') \notin \X_{x_0}$; but then $y_1^{-1}(Y') =
x_1^{î^{-1}}(f^{-1}(Y')) \notin \X_{x_0}$, and $f(x_1^î) \not\sim
f(x_0)$. For $¬$(i) $\funto$ $¬$(iii), take $(\I,\F) := (X,\X_{x_0})$,
$x_1 := x_1^î$, and reuse the proof of $¬$(i) $\funto$ $¬$(ii).
\msk
In texts about Non-Standard Analysis the infinitesimal
characterization of continuity is presented in another form:
\begin{quotation}
(i) $f$ is continuous at $x_0$
$\iff$ (iv) for all $(\I,\U)$ and $x_1 \sim x_0$, we have $f(x_1) \sim f(x_0)$.
\end{quotation}
Clearly, (iii)$\funto$(iv); but to show that (iv) implies the rest we
need to be in a universe with enough ultrafilters.
Each of the cells in the diagram in sec.\ 5 is an instance of the key
theorem --- maybe slightly disguised. For example, to prove that $g(b
+ \o) = (g'(b) + \o') \o$ we may start with $\frac{g(b + \o)}{\o} -
g'(b) = \o'$, for an infinitesimal $\o \neq 0$, i.e., $\lim_{\ee \to
0} \frac{g(b + \o)}{\o}$.
What really matters, when we look at the diagrams, is that for any
$(\I,\F)$ and for any infinitesimal $x_1: (\I,\F) \to (X,\X_{x_0})$
--- maybe obeying some condition, like $\o \neq 0$ --- there is a
unique ``adequate'' infinitesimal $y_1: (\I,\F) \to (Y,\Y_{y_0})$; we
want to ``represent'' the operation $x_1 \mapsto y_1$ as a function
$f: (X,\X_{x_0}) \to (Y,\Y_{y_0})$, and we can do that trivially by
setting $(\I,\F) := (X,\X_{x_0})$, $x_1 := x_1^î$; then we can take $f
:= y_1$, and the $f$ obtained in this way works in the general case.
%D diagram obtaining-f
%D 2Dx 100 +35 +40 +40
%D 2D 100 a0 b0
%D 2D
%D 2D +30 a1 a2 b1 b2
%D 2D
%D (( a0 .tex= \IF a1 .tex= \XXz a2 .tex= \YYz
%D b0 .tex= \XXz b1 .tex= \XXz b2 .tex= \YYz
%D a0 a1 -> .plabel= l x_1
%D a0 a2 -> .PLABEL= ^(0.61) y_1
%D a1 a2 .> .plabel= b f
%D a0 a1 midpoint a0 a2 midpoint |-> sl_
%D b0 b1 -> .plabel= l x_1^î
%D b0 b2 -> .plabel= r y_1
%D b1 b2 -> .plabel= b f:=y_1
%D ))
%D enddiagram
%D
$$\diag{obtaining-f}$$
Applying this idea to the composite of all cells in the example in
sec.\ 5, we get this:
%
%D diagram wo3
%D 2Dx 100 +25 +25 +35 +35 +40
%D 2D 100 {}i n n{}
%D 2D - / - / -
%D 2D | \ | \ |
%D 2D v v v v |
%D 2D +20 {}Ï |-> {}\o''' Ï |-----> \o''' |
%D 2D - - - - |
%D 2D | | | | |
%D 2D v v v v v
%D 2D +20 h_5 ==== h_9 h_5(Ï) == h_9(\o''') h_5(n) == h_9(n)
%D 2D
%D (( {}i {}Ï {}\o''' h_5 h_9
%D @ 0 @ 1 |-> @ 0 @ 2 |->
%D @ 1 @ 2 |-> @ 1 @ 3 |-> @ 2 @ 4 |-> @ 3 @ 4 =
%D ))
%D (( n Ï \o''' h_5(Ï) .tex= (1+\aw)^Ï h_9(\o''') .tex= e^a+\o'''
%D @ 0 @ 1 |-> @ 0 @ 2 |->
%D @ 1 @ 2 |-> @ 1 @ 3 |-> @ 2 @ 4 |-> @ 3 @ 4 =
%D ))
%D (( n{} h_5(n) .tex= (1+\aw)^n h_9(n) .tex= e^a+\o'''(n)
%D @ 0 @ 1 |-> @ 0 @ 2 |-> @ 1 @ 2 =
%D ))
%D enddiagram
%D
$$\diag{wo3}$$
%
where $i \in \IF$, $n, Ï \in \NN$, and all the other ``points'' live
in $\RRz$. Note that the `$\mto$' arrows in this diagram do not stand
for functions in the usual sense, but for functions between filtered
spaces (not necessarily total). Incidentally, all of them are
continuous.
%*
\end{document}
% Local Variables:
% coding: raw-text-unix
% ee-anchor-format: "«%s»"
% End: