% (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.







%

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