initial commit, already worked on chp 1.1, 1.2
This commit is contained in:
commit
0afce34147
SSH key fingerprint:
SHA256:G4+ddwoZmhLPRB1agvXzZMXIzkVJ36dUYZXf5NxT+u8
16 changed files with 1092 additions and 0 deletions
303
.gitignore
vendored
Normal file
303
.gitignore
vendored
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|
|
@ -0,0 +1,303 @@
|
|||
# ---> TeX
|
||||
## Core latex/pdflatex auxiliary files:
|
||||
*.aux
|
||||
*.lof
|
||||
*.log
|
||||
*.lot
|
||||
*.fls
|
||||
*.out
|
||||
*.toc
|
||||
*.fmt
|
||||
*.fot
|
||||
*.cb
|
||||
*.cb2
|
||||
.*.lb
|
||||
|
||||
## Intermediate documents:
|
||||
*.dvi
|
||||
*.xdv
|
||||
*-converted-to.*
|
||||
# these rules might exclude image files for figures etc.
|
||||
# *.ps
|
||||
# *.eps
|
||||
# *.pdf
|
||||
|
||||
## Generated if empty string is given at "Please type another file name for output:"
|
||||
*.pdf
|
||||
|
||||
## Bibliography auxiliary files (bibtex/biblatex/biber):
|
||||
*.bbl
|
||||
*.bcf
|
||||
*.blg
|
||||
*-blx.aux
|
||||
*-blx.bib
|
||||
*.run.xml
|
||||
|
||||
## Build tool auxiliary files:
|
||||
*.fdb_latexmk
|
||||
*.synctex
|
||||
*.synctex(busy)
|
||||
*.synctex.gz
|
||||
*.synctex.gz(busy)
|
||||
*.pdfsync
|
||||
|
||||
## Build tool directories for auxiliary files
|
||||
# latexrun
|
||||
latex.out/
|
||||
|
||||
## Auxiliary and intermediate files from other packages:
|
||||
# algorithms
|
||||
*.alg
|
||||
*.loa
|
||||
|
||||
# achemso
|
||||
acs-*.bib
|
||||
|
||||
# amsthm
|
||||
*.thm
|
||||
|
||||
# beamer
|
||||
*.nav
|
||||
*.pre
|
||||
*.snm
|
||||
*.vrb
|
||||
|
||||
# changes
|
||||
*.soc
|
||||
|
||||
# comment
|
||||
*.cut
|
||||
|
||||
# cprotect
|
||||
*.cpt
|
||||
|
||||
# elsarticle (documentclass of Elsevier journals)
|
||||
*.spl
|
||||
|
||||
# endnotes
|
||||
*.ent
|
||||
|
||||
# fixme
|
||||
*.lox
|
||||
|
||||
# feynmf/feynmp
|
||||
*.mf
|
||||
*.mp
|
||||
*.t[1-9]
|
||||
*.t[1-9][0-9]
|
||||
*.tfm
|
||||
|
||||
#(r)(e)ledmac/(r)(e)ledpar
|
||||
*.end
|
||||
*.?end
|
||||
*.[1-9]
|
||||
*.[1-9][0-9]
|
||||
*.[1-9][0-9][0-9]
|
||||
*.[1-9]R
|
||||
*.[1-9][0-9]R
|
||||
*.[1-9][0-9][0-9]R
|
||||
*.eledsec[1-9]
|
||||
*.eledsec[1-9]R
|
||||
*.eledsec[1-9][0-9]
|
||||
*.eledsec[1-9][0-9]R
|
||||
*.eledsec[1-9][0-9][0-9]
|
||||
*.eledsec[1-9][0-9][0-9]R
|
||||
|
||||
# glossaries
|
||||
*.acn
|
||||
*.acr
|
||||
*.glg
|
||||
*.glo
|
||||
*.gls
|
||||
*.glsdefs
|
||||
*.lzo
|
||||
*.lzs
|
||||
*.slg
|
||||
*.slo
|
||||
*.sls
|
||||
|
||||
# uncomment this for glossaries-extra (will ignore makeindex's style files!)
|
||||
# *.ist
|
||||
|
||||
# gnuplot
|
||||
*.gnuplot
|
||||
*.table
|
||||
|
||||
# gnuplottex
|
||||
*-gnuplottex-*
|
||||
|
||||
# gregoriotex
|
||||
*.gaux
|
||||
*.glog
|
||||
*.gtex
|
||||
|
||||
# htlatex
|
||||
*.4ct
|
||||
*.4tc
|
||||
*.idv
|
||||
*.lg
|
||||
*.trc
|
||||
*.xref
|
||||
|
||||
# hyperref
|
||||
*.brf
|
||||
|
||||
# knitr
|
||||
*-concordance.tex
|
||||
# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
|
||||
# *.tikz
|
||||
*-tikzDictionary
|
||||
|
||||
# listings
|
||||
*.lol
|
||||
|
||||
# luatexja-ruby
|
||||
*.ltjruby
|
||||
|
||||
# makeidx
|
||||
*.idx
|
||||
*.ilg
|
||||
*.ind
|
||||
|
||||
# minitoc
|
||||
*.maf
|
||||
*.mlf
|
||||
*.mlt
|
||||
*.mtc[0-9]*
|
||||
*.slf[0-9]*
|
||||
*.slt[0-9]*
|
||||
*.stc[0-9]*
|
||||
|
||||
# minted
|
||||
_minted*
|
||||
*.pyg
|
||||
|
||||
# morewrites
|
||||
*.mw
|
||||
|
||||
# newpax
|
||||
*.newpax
|
||||
|
||||
# nomencl
|
||||
*.nlg
|
||||
*.nlo
|
||||
*.nls
|
||||
|
||||
# pax
|
||||
*.pax
|
||||
|
||||
# pdfpcnotes
|
||||
*.pdfpc
|
||||
|
||||
# sagetex
|
||||
*.sagetex.sage
|
||||
*.sagetex.py
|
||||
*.sagetex.scmd
|
||||
|
||||
# scrwfile
|
||||
*.wrt
|
||||
|
||||
# svg
|
||||
svg-inkscape/
|
||||
|
||||
# sympy
|
||||
*.sout
|
||||
*.sympy
|
||||
sympy-plots-for-*.tex/
|
||||
|
||||
# pdfcomment
|
||||
*.upa
|
||||
*.upb
|
||||
|
||||
# pythontex
|
||||
*.pytxcode
|
||||
pythontex-files-*/
|
||||
|
||||
# tcolorbox
|
||||
*.listing
|
||||
|
||||
# thmtools
|
||||
*.loe
|
||||
|
||||
# TikZ & PGF
|
||||
*.dpth
|
||||
*.md5
|
||||
*.auxlock
|
||||
|
||||
# titletoc
|
||||
*.ptc
|
||||
|
||||
# todonotes
|
||||
*.tdo
|
||||
|
||||
# vhistory
|
||||
*.hst
|
||||
*.ver
|
||||
|
||||
# easy-todo
|
||||
*.lod
|
||||
|
||||
# xcolor
|
||||
*.xcp
|
||||
|
||||
# xmpincl
|
||||
*.xmpi
|
||||
|
||||
# xindy
|
||||
*.xdy
|
||||
|
||||
# xypic precompiled matrices and outlines
|
||||
*.xyc
|
||||
*.xyd
|
||||
|
||||
# endfloat
|
||||
*.ttt
|
||||
*.fff
|
||||
|
||||
# Latexian
|
||||
TSWLatexianTemp*
|
||||
|
||||
## Editors:
|
||||
# WinEdt
|
||||
*.bak
|
||||
*.sav
|
||||
|
||||
# Texpad
|
||||
.texpadtmp
|
||||
|
||||
# LyX
|
||||
*.lyx~
|
||||
|
||||
# Kile
|
||||
*.backup
|
||||
|
||||
# gummi
|
||||
.*.swp
|
||||
|
||||
# KBibTeX
|
||||
*~[0-9]*
|
||||
|
||||
# TeXnicCenter
|
||||
*.tps
|
||||
|
||||
# auto folder when using emacs and auctex
|
||||
./auto/*
|
||||
*.el
|
||||
|
||||
# expex forward references with \gathertags
|
||||
*-tags.tex
|
||||
|
||||
# standalone packages
|
||||
*.sta
|
||||
|
||||
# Makeindex log files
|
||||
*.lpz
|
||||
|
||||
# xwatermark package
|
||||
*.xwm
|
||||
|
||||
# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
|
||||
# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
|
||||
# Uncomment the next line to have this generated file ignored.
|
||||
#*Notes.bib
|
||||
|
||||
9
LICENSE
Normal file
9
LICENSE
Normal file
|
|
@ -0,0 +1,9 @@
|
|||
MIT License
|
||||
|
||||
Copyright (c) 2024 leonv
|
||||
|
||||
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
|
||||
|
||||
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
|
||||
|
||||
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
|
||||
2
README.md
Normal file
2
README.md
Normal file
|
|
@ -0,0 +1,2 @@
|
|||
# Topologie
|
||||
Meine Zusammenfassung des Moduls "Topologie" gehalten im SS24 von Kang Li.
|
||||
4
tex/.chktexrc
Normal file
4
tex/.chktexrc
Normal file
|
|
@ -0,0 +1,4 @@
|
|||
CmdLine = {
|
||||
-n18
|
||||
-n46
|
||||
}
|
||||
3
tex/.vscode/ltex.dictionary.de-DE.txt
vendored
Normal file
3
tex/.vscode/ltex.dictionary.de-DE.txt
vendored
Normal file
|
|
@ -0,0 +1,3 @@
|
|||
Kofinite
|
||||
Koabzählbare
|
||||
kofinite
|
||||
1
tex/.vscode/ltex.hiddenFalsePositives.de-DE.txt
vendored
Normal file
1
tex/.vscode/ltex.hiddenFalsePositives.de-DE.txt
vendored
Normal file
|
|
@ -0,0 +1 @@
|
|||
{"rule":"DE_CASE","sentence":"^\\Q[(T3)] Endliche Schnitte von Mengen in \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q sind in \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q enthalten.\\E$"}
|
||||
38
tex/.vscode/settings.json
vendored
Normal file
38
tex/.vscode/settings.json
vendored
Normal file
|
|
@ -0,0 +1,38 @@
|
|||
{
|
||||
"latex-workshop.latex.tools": [
|
||||
{
|
||||
"name": "latexmk-main",
|
||||
"command": "latexmk",
|
||||
"args": [
|
||||
"-synctex=1",
|
||||
"-interaction=nonstopmode",
|
||||
"-file-line-error",
|
||||
"-shell-escape",
|
||||
"-pdf",
|
||||
"-xelatex",
|
||||
"-outdir=%OUTDIR%",
|
||||
"main.tex"
|
||||
],
|
||||
"env": {}
|
||||
}
|
||||
],
|
||||
"latex-workshop.latex.recipes": [
|
||||
{
|
||||
"name": "latexmk-main",
|
||||
"tools": [
|
||||
"latexmk-main"
|
||||
]
|
||||
}
|
||||
],
|
||||
"ltex.latex.commands": {
|
||||
"\\customlabel{}": "ignore",
|
||||
"\\setminted[]{}": "ignore",
|
||||
"\\setmintedinline[]{}": "ignore",
|
||||
"\\setmathfont[]{}": "ignore",
|
||||
"\\setmathfont{}": "ignore"
|
||||
},
|
||||
"ltex.latex.environments": {
|
||||
"cases": "ignore"
|
||||
},
|
||||
"ltex.language": "de-DE"
|
||||
}
|
||||
15
tex/Makefile
Normal file
15
tex/Makefile
Normal file
|
|
@ -0,0 +1,15 @@
|
|||
src = $(wildcard *.tex)
|
||||
pdf = $(src:.tex=.pdf)
|
||||
|
||||
.PHONY: all clean
|
||||
|
||||
all: $(pdf)
|
||||
|
||||
%.pdf: %.tex $(wildcard src/*.tex) $(wildcard *.bib)
|
||||
latexmk -pdf -xelatex -shell-escape -file-line-error -synctex=1 -halt-on-error -shell-escape $<
|
||||
|
||||
clean:
|
||||
latexmk -C $(src)
|
||||
rm -f $(wildcard *.out *.nls *.nlo *.bbl *.blg *-blx.bib *.run.xml *.bcf *.synctex.gz *.fdb_latexmk *.fls *.toc *.loe *.tdo *.bbl-SAVE-ERROR)
|
||||
rm -f $(wildcard sections/*.aux)
|
||||
rm -rf $(wildcard _minted-main sections/.auctex-auto _region_.prv)
|
||||
17
tex/bib.bib
Normal file
17
tex/bib.bib
Normal file
|
|
@ -0,0 +1,17 @@
|
|||
@article{poll1999algebra,
|
||||
title = {Algebra of Programming by Richard Bird and Oege de Moor, Prentice Hall, 1996 (dated 1997).},
|
||||
author = {Poll, Erik and Thompson, Simon},
|
||||
journal = {Journal of Functional Programming},
|
||||
volume = {9},
|
||||
number = {3},
|
||||
pages = {347--354},
|
||||
year = {1999},
|
||||
publisher = {Cambridge University Press}
|
||||
}
|
||||
|
||||
@book{adamek1990abstract,
|
||||
title = {Abstract and concrete categories},
|
||||
author = {Ad{\'a}mek, Ji{\v{r}}{\'\i} and Herrlich, Horst and Strecker, George},
|
||||
year = {1990},
|
||||
publisher = {Wiley-Interscience}
|
||||
}
|
||||
246
tex/catprog.sty
Normal file
246
tex/catprog.sty
Normal file
|
|
@ -0,0 +1,246 @@
|
|||
%
|
||||
% Standard macros to typeset papers on category theory and semantics
|
||||
%
|
||||
% Unless \catname is defined, make it bold.
|
||||
% So, call \providecommand{\catname}{\mathcal}
|
||||
% or \reprovidecommand{\catname}{\mathcal}
|
||||
% before calling this file if you prefer calligraphic names for categories.
|
||||
%
|
||||
% Same applies to other commands.
|
||||
%
|
||||
% COPIED FROM https://gitlab.cs.fau.de/i8/TexCommon/ AND THEN ADJUSTED.
|
||||
|
||||
\NeedsTeXFormat{LaTeX2e}
|
||||
\ProvidesPackage{catprog}[2024/03/30 Macros for Category Theory and Semantics]
|
||||
|
||||
\RequirePackage{bm} % Needed for defbbname
|
||||
\RequirePackage{mathtools} % Needed for coloneqq
|
||||
|
||||
\providecommand{\catname}{\mathsf}
|
||||
%\providecommand{\mndname}{\mathbbb}
|
||||
\providecommand{\clsname}{\mathcal}
|
||||
% \providecommand{\clsname}{\mathscr}
|
||||
\providecommand{\oname}[1]{\operatorname{\mathsf{#1}}}
|
||||
|
||||
%% Defining category names like \BA, \BB, etc
|
||||
\def\defcatname#1{\expandafter\def\csname B#1\endcsname{\catname{#1}}}
|
||||
\def\defcatnames#1{\ifx#1\defcatnames\else\defcatname#1\expandafter\defcatnames\fi}
|
||||
\defcatnames ABCDEFGHIJKLMNOPQRSTUVWXYZ\defcatnames
|
||||
|
||||
%% Defining \CA, \CB, etc
|
||||
\def\defclsname#1{\expandafter\def\csname C#1\endcsname{\clsname{#1}}}
|
||||
\def\defclsnames#1{\ifx#1\defclsnames\else\defclsname#1\expandafter\defclsnames\fi}
|
||||
\defclsnames ABCDEFGHIJKLMNOPQRSTUVWXYZ\defclsnames
|
||||
|
||||
%% Defining \BBA, \BBB, etc
|
||||
\def\defbbname#1{\expandafter\def\csname BB#1\endcsname{{\bm{\mathsf{#1}}}}}
|
||||
\def\defbbnames#1{\ifx#1\defbbnames\else\defbbname#1\expandafter\defbbnames\fi}
|
||||
\defbbnames ABCDEFGHIJKLMNOPQRSTUVWXYZ\defbbnames
|
||||
|
||||
%% Defining \BMA, \BMB, etc
|
||||
\def\defbbname#1{\expandafter\def\csname BM#1\endcsname{{\bm{\mathsf{#1}}}}}
|
||||
\def\defbbnames#1{\ifx#1\defbbnames\else\defbbname#1\expandafter\defbbnames\fi}
|
||||
\defbbnames ABCDEFGHIJKLMNOPQRSTUVWXYZ\defbbnames
|
||||
|
||||
%% Some standard categories
|
||||
\def\Set{\ensuremath{\catname{Set}}}
|
||||
\def\Par{\ensuremath{\catname{Par}}}
|
||||
\def\Rel{\ensuremath{\catname{Rel}}}
|
||||
\def\Cpo{\ensuremath{\catname{Cpo}}}
|
||||
\def\Pos{\ensuremath{\catname{Pos}}}
|
||||
\def\Mon{\ensuremath{\catname{Mon}}}
|
||||
\def\Gra{\ensuremath{\catname{Gra}}}
|
||||
\providecommand{\Alg}[1]{\ensuremath{\catname{Alg}(#1)}}
|
||||
\providecommand{\Coalg}[1]{\ensuremath{\catname{Coalg}(#1)}}
|
||||
%% Objects of category
|
||||
\providecommand{\obj}[1]{\ensuremath{\vert #1 \vert}}
|
||||
%% Dual category
|
||||
\providecommand{\dual}[1]{\ensuremath{#1^{op}}}
|
||||
\providecommand{\ite}[3]{\oname{if}#1\oname{then}#2\oname{else}}
|
||||
|
||||
% Misc
|
||||
%% Commutation of diagrams
|
||||
\providecommand{\comm}{\circlearrowleft}
|
||||
%% Banana brackets for catamorphisms
|
||||
\RequirePackage{stmaryrd}
|
||||
\providecommand{\cata}[1]{\llparenthesis #1 \rrparenthesis}
|
||||
% Lens brackets for anamorphisms https://unicodeplus.com/U+3016
|
||||
\providecommand{\ana}[1]{〖 #1 〗}
|
||||
\providecommand{\dom}{\oname{dom}}
|
||||
|
||||
\providecommand{\eps}{{\operatorname\epsilon}}
|
||||
\providecommand{\amp}{\mathbin{\&}}
|
||||
|
||||
\providecommand{\argument}{\operatorname{-\!-}}
|
||||
\providecommand{\altargument}{\underline{\;\;}}
|
||||
|
||||
\providecommand{\wave}[1]{\widetilde{#1}} % Overline wave
|
||||
\providecommand{\ul}{\underline} % Underline
|
||||
|
||||
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
|
||||
\providecommand{\mplus}{{\scriptscriptstyle\bf+}} % Small '+' for indexing
|
||||
|
||||
\providecommand{\smin}{\smallsetminus}
|
||||
|
||||
\providecommand{\ass}{\mathrel{\coloneqq}}
|
||||
\providecommand{\bnf}{\mathrel{\Coloneqq}}
|
||||
|
||||
%% Some standard functorsw
|
||||
\providecommand{\PSet}{{\mathcal P}} % Powerset
|
||||
\providecommand{\FSet}{{\mathcal P}_{\omega}} % Finite powerset
|
||||
\providecommand{\CSet}{{\mathcal P}_{\omega_1}} % Countable powerset
|
||||
\providecommand{\NESet}{{\mathcal P}^{\mplus}} % Non-empty powerset
|
||||
\providecommand{\Id}{\operatorname{Id}}
|
||||
|
||||
%% General categorical notation
|
||||
\providecommand{\Hom}{\mathsf{Hom}}
|
||||
\providecommand{\id}{\mathsf{id}}
|
||||
\providecommand{\op}{\mathsf{op}}
|
||||
\providecommand{\comp}{\mathbin{\circ}}
|
||||
\providecommand{\iso}{\mathbin{\cong}}
|
||||
\providecommand{\tensor}{\mathbin{\otimes}}
|
||||
\providecommand{\unit}{\star}
|
||||
\providecommand{\bang}{\operatorname!} % Terminal map
|
||||
\providecommand{\cobang}{\operatorname¡} % Initial map
|
||||
|
||||
%% Various arrows
|
||||
\providecommand{\from}{\leftarrow}
|
||||
|
||||
\providecommand{\ito}{\hookrightarrow} % Injection
|
||||
\providecommand{\ifrom}{\hookleftarrow}
|
||||
|
||||
\providecommand{\pto}{\mathrel{\rightharpoonup}} % Partial function
|
||||
\providecommand{\pfrom}{\mathrel{\leftarpoonup}} %
|
||||
|
||||
\providecommand{\tto}{\mathrel{\Rightarrow}} % Double arrow
|
||||
\providecommand{\tfrom}{\mathrel{\Leftarrow}} %
|
||||
|
||||
\providecommand{\mto}{\mapsto}
|
||||
|
||||
\providecommand{\xto}[1]{\,\xrightarrow{#1}\,}
|
||||
\providecommand{\xfrom}[1]{\,\xleftarrow{\;#1}\,}
|
||||
|
||||
\providecommand{\To}{\mathrel{\Rightarrow}} % Double arrow
|
||||
\providecommand{\From}{\mathrel{\Leftarrow}}
|
||||
|
||||
\providecommand{\dar}{\kern-.75pt\operatorname{\downarrow}}
|
||||
\providecommand{\uar}{\kern-.75pt\operatorname{\uparrow}}
|
||||
|
||||
\providecommand{\Dar}{\kern-.75pt\operatorname{\Downarrow}}
|
||||
\providecommand{\Uar}{\kern-.75pt\operatorname{\Uparrow}}
|
||||
|
||||
|
||||
%% Logic
|
||||
\providecommand{\True}{\top}
|
||||
\providecommand{\False}{\bot}
|
||||
|
||||
\providecommand{\bigor}{\bigvee}
|
||||
\providecommand{\bigand}{\bigwedge}
|
||||
|
||||
\providecommand{\impl}{\Rightarrow}
|
||||
\providecommand{\equ}{\Longleftrightarrow}
|
||||
|
||||
\providecommand{\entails}{\vdash}
|
||||
|
||||
%% Order
|
||||
\providecommand{\appr}{\sqsubseteq}
|
||||
\providecommand{\join}{\sqcup}
|
||||
\providecommand{\meet}{\sqcap}
|
||||
|
||||
\providecommand{\bigjoin}{\bigsqcup}
|
||||
\providecommand{\bigmeet}{\bigsqcap}
|
||||
|
||||
%% Products
|
||||
\providecommand{\fst}{\pi_1}
|
||||
\providecommand{\snd}{\pi_2}
|
||||
\providecommand{\pr}{\oname{pr}}
|
||||
|
||||
\providecommand{\brks}[1]{\langle #1\rangle}
|
||||
\providecommand{\Brks}[1]{\bigl\langle #1\bigr\rangle}
|
||||
|
||||
%% Coproducts
|
||||
\providecommand{\inl}{\oname{i}_1}
|
||||
\providecommand{\inr}{\oname{i}_2}
|
||||
\providecommand{\inj}{\oname{in}}
|
||||
|
||||
\DeclareSymbolFont{Symbols}{OMS}{cmsy}{m}{n}
|
||||
\DeclareMathSymbol{\iobj}{\mathord}{Symbols}{"3B}
|
||||
%\DeclareRobustCommand{\iobj}{\emptyset}
|
||||
|
||||
%% CCC
|
||||
\providecommand{\curry}{\oname{curry}}
|
||||
\providecommand{\uncurry}{\oname{uncurry}}
|
||||
\providecommand{\ev}{\oname{ev}}
|
||||
|
||||
% Semantic brackets
|
||||
\RequirePackage{stmaryrd}
|
||||
|
||||
\providecommand{\lsem}{\llbracket}
|
||||
\providecommand{\rsem}{\rrbracket}
|
||||
\providecommand{\sem}[1]{\lsem #1 \rsem}
|
||||
|
||||
\providecommand{\Lsem}{\bigl\llbracket}
|
||||
\providecommand{\Rsem}{\bigr\rrbracket}
|
||||
\providecommand{\Sem}[1]{\Lsem #1 \Rsem}
|
||||
|
||||
% Typographic
|
||||
\providecommand{\comma}{,\operatorname{}\linebreak[1]} % possibly line-breaking comma
|
||||
\providecommand{\dash}{\nobreakdash-\hspace{0pt}} % non-line-breaking hyphen
|
||||
\providecommand{\erule}{\rule{0pt}{0pt}} % Empty object whose emptiness
|
||||
% is not detected by LaTeX
|
||||
|
||||
\providecommand{\by}[1]{\text{/\!\!/~#1}} % Comments in equations
|
||||
\providecommand{\pacman}[1]{} % Hide a piece of text
|
||||
|
||||
\newcommand{\undefine}[1]{\let #1\relax} % Make a command undefined
|
||||
|
||||
\providecommand{\noqed}{\def\qed{}} % Undefine the QED symbol
|
||||
|
||||
% -1 superscript for the inversion operator
|
||||
\providecommand{\mone}{{\text{\kern.5pt\rmfamily-}\mathsf{\kern-.5pt1}}}
|
||||
|
||||
\makeatletter
|
||||
\@ifpackageloaded{enumitem}{}{
|
||||
\RequirePackage[loadonly]{enumitem} % without [loadonly]
|
||||
} % conflicts with Beamer
|
||||
\makeatother
|
||||
|
||||
% Condensed list environments
|
||||
\newlist{citemize}{itemize}{1}
|
||||
\setlist[citemize]{label=\labelitemi,wide}
|
||||
%leftmargin=0cm,itemindent=.7cm,labelwidth=\itemindent,labelsep=-.3cm,align=left}
|
||||
|
||||
\newlist{cenumerate}{enumerate}{1}
|
||||
\setlist[cenumerate,1]{label=\arabic*.~,ref={\arabic*},wide}
|
||||
%leftmargin=0cm,itemindent=.7cm,labelwidth=\itemindent,labelsep=-.3cm,align=left}
|
||||
|
||||
%\newenvironment{citemize}{\begin{itemize}[leftmargin=0cm,itemindent=.7cm,labelwidth=\itemindent,labelsep=-.3cm,align=left]}{\end{itemize}}
|
||||
%\newenvironment{cenumerate}{\begin{enumerate}[leftmargin=0cm,itemindent=.7cm,labelwidth=\itemindent,labelsep=-.3cm,align=left]}{\end{enumerate}}
|
||||
|
||||
%% A macro for defining mixfix operators
|
||||
\makeatletter
|
||||
\def\mfix#1{\oname{#1}\@ifnextchar\bgroup\@mfix{}} % processing odd arguments
|
||||
\def\@mfix#1{#1\@ifnextchar\bgroup\mfix{}} % processing even arguments
|
||||
\makeatother
|
||||
|
||||
% %% Instances if mfix
|
||||
% \providecommand{\ift}[3]{\mfix{if}{\mathbin{}#1}{then}{\mathbin{}#2}{else}{\mathbin{}#3}}
|
||||
% \providecommand{\case}[3]{\mfix{case}{\mathbin{}#1}{of}{#2}{\kern-1pt;}{\mathbin{}#3}}
|
||||
|
||||
%%% Overwrite greek letters with var versions https://tex.stackexchange.com/a/304576/292879
|
||||
\AfterBeginDocument{
|
||||
\renewcommand{\epsilon}{\varepsilon}
|
||||
\renewcommand{\theta}{\vartheta}
|
||||
\renewcommand{\theta}{\vartheta}
|
||||
\renewcommand{\kappa}{\varkappa}
|
||||
\renewcommand{\rho}{\varrho}
|
||||
\renewcommand{\phi}{\varphi}
|
||||
}
|
||||
%%%
|
||||
|
||||
%%% Topology
|
||||
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
|
||||
\newcommand{\inter}[1]{\mathring{#1}}
|
||||
\newcommand{\clos}[1]{\overline{#1}}
|
||||
\newcommand{\bound}[1]{\partial#1}
|
||||
%%%
|
||||
153
tex/main.tex
Normal file
153
tex/main.tex
Normal file
|
|
@ -0,0 +1,153 @@
|
|||
\documentclass[a4paper,11pt,titlepage,numbers=noenddot,ngerman]{scrbook}
|
||||
|
||||
%%%% Document Setup
|
||||
\usepackage[top=2cm,lmargin=1in,rmargin=1in,bottom=3cm,hmarginratio=1:1]{geometry}
|
||||
\usepackage{scrlayer-scrpage}
|
||||
\usepackage[ngerman]{babel}
|
||||
\babeltags{german=ngerman}
|
||||
\usepackage{anyfontsize}
|
||||
\usepackage{mathtools}
|
||||
\usepackage[warnings-off={mathtools-colon,mathtools-overbracket}]{unicode-math}
|
||||
\setmathfont[math-style=ISO,version=normal]{Latin Modern Math}
|
||||
\setmathfont[version=bold,math-style=ISO,FakeBold = 3]{Latin Modern Math}
|
||||
\setmathfont[range={\mathcal,\mathbfcal},StylisticSet=1]{XITS Math}
|
||||
\setmathfont[range=\mathscr]{XITS Math}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{stmaryrd}
|
||||
% \usepackage{mathrsfs}
|
||||
\usepackage{booktabs}
|
||||
\usepackage[scale=.85]{noto-mono} % TODO find better unicode mono font
|
||||
\usepackage[final]{hyperref}
|
||||
\usepackage{scrhack}
|
||||
\automark{chapter}
|
||||
\renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}
|
||||
\renewcommand{\subsectionmark}[1]{\markright{\thesubsection\ #1}}
|
||||
\pagestyle{scrheadings}
|
||||
\newcounter{resumeenum} % for resuming enumerated lists, https://tex.stackexchange.com/a/1702
|
||||
\usepackage{catprog}
|
||||
\usepackage{multicol}
|
||||
\usepackage[moderate, title=tight, bibliography=tight, margins=tight]{savetrees}
|
||||
%%%%
|
||||
|
||||
%%%% Metadata
|
||||
\newcommand{\DocTitle}{Zusammenfassung}
|
||||
\newcommand{\UniCourse}{(Mengentheoretische) Topologie}
|
||||
\newcommand{\UniProf}{Dr.\ Kang Li}
|
||||
\author{Leon Vatthauer}
|
||||
%%%%
|
||||
|
||||
%%%% Title
|
||||
\title{
|
||||
\vspace{2in}
|
||||
\textmd{\textbf{\UniCourse\\\DocTitle}}\\
|
||||
\vspace{0.1in}\large{Gehalten von \textit{\UniProf\ }}
|
||||
\vspace{3in}
|
||||
}
|
||||
%%%%
|
||||
|
||||
%%%% Spacing
|
||||
% \setlength{\parindent}{0pt}
|
||||
% \setlength{\parskip}{6pt}
|
||||
% \setlength{\marginparsep}{0cm}
|
||||
%%%%
|
||||
|
||||
%%%% Math packages
|
||||
\usepackage{amsthm}
|
||||
\usepackage{thmtools}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{cd, babel, quotes}
|
||||
\usepackage{quiver}
|
||||
\declaretheorem[name=Definition,style=definition,numberwithin=section]{definition}
|
||||
\declaretheorem[name=Beispiel,style=definition,sibling=definition]{example}
|
||||
\declaretheorem[style=definition,numbered=no]{exercise}
|
||||
\declaretheorem[name=Anmerkung,style=definition,sibling=definition]{remark}
|
||||
\declaretheorem[name=Annahme,style=definition,sibling=definition]{assumption}
|
||||
\declaretheorem[name=Beobachtung,style=definition,sibling=definition]{observation}
|
||||
\declaretheorem[name=Satz,sibling=definition]{theorem}
|
||||
\declaretheorem[sibling=definition]{corollary}
|
||||
\declaretheorem[name=FaktFact,sibling=definition]{fact}
|
||||
\declaretheorem[sibling=definition]{lemma}
|
||||
\declaretheorem[sibling=lemma]{proposition}
|
||||
%%%%
|
||||
|
||||
%%%% href
|
||||
\makeatletter
|
||||
\hypersetup{
|
||||
pdfauthor={\@author},
|
||||
pdftitle={\UniCourse},
|
||||
% kill those ugly red rectangles around links
|
||||
hidelinks,
|
||||
}
|
||||
\makeatother
|
||||
%%%%
|
||||
|
||||
%%%% Bibliography
|
||||
\usepackage[style=ieee, sorting=ynt, language=ngerman]{biblatex} % advanced citations, british to make dates DD-MM-YYYY
|
||||
\usepackage{csquotes} % biblatex recommended to load this
|
||||
\addbibresource{bib.bib}
|
||||
%%%%
|
||||
|
||||
%%%% Custom definitions: Commands and environments
|
||||
|
||||
%%% Case distinction environment
|
||||
% Defines the `mycase` environment, copied from https://tex.stackexchange.com/questions/251053/how-to-use-case-1-case-2-in-a-proof-ieee-confs
|
||||
\newcounter{cases}
|
||||
\newcounter{subcases}[cases]
|
||||
\newenvironment{mycase}
|
||||
{
|
||||
\setcounter{cases}{0}
|
||||
\setcounter{subcases}{0}
|
||||
\newcommand{\case}
|
||||
{
|
||||
\par\indent\stepcounter{cases}\textbf{Fall \thecases.}
|
||||
}
|
||||
\newcommand{\subcase}
|
||||
{
|
||||
\par\indent\stepcounter{subcases}\textit{Fall (\thesubcases):}
|
||||
}
|
||||
}
|
||||
{
|
||||
\par
|
||||
}
|
||||
\renewcommand*\thecases{\arabic{cases}}
|
||||
\renewcommand*\thesubcases{\roman{subcases}}
|
||||
%%%
|
||||
|
||||
%%% Custom label command
|
||||
\makeatletter
|
||||
\newcommand{\customlabel}[2]{%
|
||||
\protected@write \@auxout {}{\string \newlabel {#1}{{#2}{\thepage}{#2}{#1}{}} }% chktex 1
|
||||
\hypertarget{#1}{#2}%
|
||||
}
|
||||
\makeatother
|
||||
%%%
|
||||
|
||||
%%% Unicode substitutions
|
||||
\def\circlearrowleft{↺}
|
||||
%%%
|
||||
%%%%
|
||||
|
||||
\begin{document}
|
||||
\selectlanguage{ngerman}
|
||||
%% Titlepage
|
||||
\maketitle
|
||||
\thispagestyle{empty}
|
||||
\null\newpage
|
||||
\setcounter{page}{1}
|
||||
%%
|
||||
|
||||
%% TOC
|
||||
\tableofcontents
|
||||
%%
|
||||
|
||||
%% Contents
|
||||
\include{sections/01_topräume}
|
||||
\include{sections/02_zusammenhang}
|
||||
\include{sections/03_konstruktion}
|
||||
\include{sections/04_kompaktheit}
|
||||
%%
|
||||
|
||||
\appendix
|
||||
\emergencystretch=1em
|
||||
\printbibliography[heading=bibintoc]{}
|
||||
\end{document}
|
||||
40
tex/quiver.sty
Normal file
40
tex/quiver.sty
Normal file
|
|
@ -0,0 +1,40 @@
|
|||
% *** quiver ***
|
||||
% A package for drawing commutative diagrams exported from https://q.uiver.app.
|
||||
%
|
||||
% This package is currently a wrapper around the `tikz-cd` package, importing necessary TikZ
|
||||
% libraries, and defining a new TikZ style for curves of a fixed height.
|
||||
%
|
||||
% Version: 1.4.2
|
||||
% Authors:
|
||||
% - varkor (https://github.com/varkor)
|
||||
% - AndréC (https://tex.stackexchange.com/users/138900/andr%C3%A9c)
|
||||
|
||||
\NeedsTeXFormat{LaTeX2e}
|
||||
\ProvidesPackage{quiver}[2021/01/11 quiver]
|
||||
|
||||
% `tikz-cd` is necessary to draw commutative diagrams.
|
||||
\RequirePackage{tikz-cd}
|
||||
% `amssymb` is necessary for `\lrcorner` and `\ulcorner`.
|
||||
% \RequirePackage{amssymb}
|
||||
% `calc` is necessary to draw curved arrows.
|
||||
\usetikzlibrary{calc}
|
||||
% `pathmorphing` is necessary to draw squiggly arrows.
|
||||
\usetikzlibrary{decorations.pathmorphing}
|
||||
|
||||
% A TikZ style for curved arrows of a fixed height, due to AndréC.
|
||||
\tikzset{curve/.style={settings={#1},to path={(\tikztostart)
|
||||
.. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
|
||||
and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
|
||||
.. (\tikztotarget)\tikztonodes}},
|
||||
settings/.code={\tikzset{quiver/.cd,#1}
|
||||
\def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}},
|
||||
quiver/.cd,pos/.initial=0.35,height/.initial=0}
|
||||
|
||||
% TikZ arrowhead/tail styles.
|
||||
\tikzset{tail reversed/.code={\pgfsetarrowsstart{tikzcd to}}}
|
||||
\tikzset{2tail/.code={\pgfsetarrowsstart{Implies[reversed]}}}
|
||||
\tikzset{2tail reversed/.code={\pgfsetarrowsstart{Implies}}}
|
||||
% TikZ arrow styles.
|
||||
\tikzset{no body/.style={/tikz/dash pattern=on 0 off 1mm}}
|
||||
|
||||
\endinput
|
||||
258
tex/sections/01_topräume.tex
Normal file
258
tex/sections/01_topräume.tex
Normal file
|
|
@ -0,0 +1,258 @@
|
|||
\chapter{Topologische Räume}\label{chp:topologies}
|
||||
\section{Topologien}
|
||||
\begin{definition}[Topologie]
|
||||
Sei $X$ eine Menge. Eine \emph{Topologie} auf $X$ ist eine Teilmenge $\CO \subseteq \CP(X)$ mit folgenden Axiomen:
|
||||
\begin{enumerate}
|
||||
\item $\emptyset, X \in \CO$.
|
||||
\item Vereinigungen von Mengen in $\CO$ sind in $\CO$ enthalten.
|
||||
\item Endliche Schnitte von Mengen in $\CO$ sind in $\CO$ enthalten.
|
||||
\end{enumerate}
|
||||
Die Mengen $O \in \CO$ heißen \emph{offen} und die Mengen $A \in X \setminus \CO$ heißen \emph{abgeschlossen}.
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}
|
||||
Topologien lassen sich auch durch abgeschlossene Mengen charakterisieren, wir erhalten folgenden Definition:
|
||||
Eine Topologie auf $X$ ist eine Teilmenge $\CO \subseteq \CP(X)$, sodass folgende Axiome gelten:
|
||||
\begin{enumerate}
|
||||
\item $\emptyset$ und $X$ sind abgeschlossen.
|
||||
\item Schnitte von abgeschlossenen Mengen sind abgeschlossen.
|
||||
\item Endliche Vereinigungen von abgeschlossenen Mengen sind abgeschlossen.
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
|
||||
\subsection{Die Diskrete und die Indiskrete Topologie}
|
||||
\begin{example}\label{ex:discrete_top}
|
||||
Für jede Menge $X$ nennen wir $\CO_{dsk} := \CP(X)$ die \emph{diskrete Topologie} auf $X$.
|
||||
Jede Teilmenge von $X$ ist offen und abgeschlossen bzgl.\ dieser Topologie.
|
||||
\end{example}
|
||||
|
||||
\begin{example}
|
||||
Für jede Menge $X$ nennen wir $\CO_{in} := \{\emptyset, X\}$ die \emph{indiskrete Topologie} auf $X$.
|
||||
\end{example}
|
||||
|
||||
\subsection{Die Kofinite und die Koabzählbare Topologie}
|
||||
\begin{example}
|
||||
Für jede Menge $X$ ist $\CO_{kof} := \{O \subseteq X \;\vert\; X \setminus O \text{ endlich, oder } O = \emptyset\}$ die \emph{kofinite Topologie} auf $X$.
|
||||
\end{example}
|
||||
|
||||
\begin{example}
|
||||
Für jede Menge $X$ ist $\CO_{koab} := \{O \subseteq X \;\vert\; X \setminus O \text{ abzählbar, oder } O = \emptyset\}$ die \emph{Koabzählbare Topologie} auf $X$.
|
||||
\end{example}
|
||||
|
||||
\subsection{Die Leere und die Einpunkttopologie}
|
||||
\begin{example}
|
||||
Auf der leeren Menge $\emptyset$ gibt es genau eine Topologie, nämlich $\CO_\emptyset := \{\emptyset\}$, die \emph{leere Topologie}.
|
||||
\end{example}
|
||||
|
||||
\begin{example}
|
||||
Auf einer einelementigen Menge $\{x\}$ gibt es ebenfalls genau eine Topologie, nämlich $\CO_{\{x\}} := \{\emptyset, \{x\}\} = \CO_{dsk} = \CO_{in}$, der \emph{Einpunktraum}.
|
||||
\end{example}
|
||||
|
||||
\subsection{Die Teilraumtopologie}
|
||||
\begin{example}
|
||||
Sei $(X, \CO_X)$ ein topologischer Raum und $M \subseteq X$ eine Teilmenge.
|
||||
Dann erhalten wir eine Topologie auf $M$:
|
||||
\[\CO_{M\subseteq X} := \CO_X \cap M = \{O \cap M \;\vert\; O \in \CO_X\}\]
|
||||
die \emph{Teilraumtopologie}.
|
||||
Offene (abgeschlossene) Mengen in $\CO_{M\subseteq X}$ sind Schnitte offener (abgeschlossener) Mengen in $X$ mit $M$.
|
||||
\end{example}
|
||||
|
||||
\begin{lemma}
|
||||
\begin{enumerate}
|
||||
\item Ist $M \subseteq X$ offen, so sind alle Teilmengen $A \subseteq M$ die offen bezüglich $\CO_{M\subseteq X}$ sind, auch offen bezüglich $\CO_X$.
|
||||
\item Ist $M \subseteq X$ abgeschlossen, so sind alle Teilmengen $A \subseteq M$ die abgeschlossen bezüglich $\CO_{M\subseteq X}$ sind, auch abgeschlossen bezüglich $\CO_X$.
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
|
||||
% TODO Praesenzblatt 1 A2c
|
||||
|
||||
\subsection{Die Metrische Topologie}
|
||||
\begin{definition}[Metrischer Raum]
|
||||
Ein \emph{metrischer Raum} ist ein Paar $(X,d)$ aus einer Menge $X$ und einer \emph{Metrik} $d : X \times X \to \mathbb{R}$, sodass:
|
||||
\begin{enumerate}
|
||||
\item \emph{Positivität}: $d(x,y) \geq 0$ und $d(x,y) = 0 \iff x = y$.
|
||||
\item \emph{Symmetrie}: $d(x,y) = d(y,x)$.
|
||||
\item \emph{Dreiecksungleichung}: $d(x,y) \leq d(x,y) + d(y,z)$.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Metrische Topologie]
|
||||
Sei $(X,d)$ ein metrischer Raum.
|
||||
\begin{enumerate}
|
||||
\item Für $x \in X$ und $r > 0$ definieren wir die \emph{offene} und die \emph{abgeschlossene} Kugel um $x$:
|
||||
\[B_r(x) = \{y \in X \;\vert\; d(x,y) < r\} \qquad B_{\leq r}(x) = \{y \in X \;\vert\; d(x,y) \leq r\}\]
|
||||
\item Die \emph{metrische Topologie} auf $X$ ist definiert durch:
|
||||
\[\CO_d := \{O \subseteq X \;\vert\; \forall x \in O. \exists \epsilon > 0. B_\epsilon(x) \subseteq O\}.\]
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
Sei $(X,d)$ ein metrischer Raum, dann gilt:
|
||||
\begin{enumerate}
|
||||
\item Die metrische Topologie ist eine Topologie auf $X$.
|
||||
\item Jedes $B_\epsilon(x)$ ist offen.
|
||||
\item Jedes $B_{\leq \epsilon}(x)$ ist abgeschlossen.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{example}[Diskrete Metrik]
|
||||
Die \emph{diskrete Metrik} auf $X$ ist definiert durch
|
||||
\[d : X \times X \to \mathbb{R}, \qquad d(x,y) = \begin{cases}
|
||||
0 & x = y \\
|
||||
1 & x \not= y
|
||||
\end{cases}\]
|
||||
Die davon induzierte Topologie ist die diskrete Topologie (\autoref{ex:discrete_top}).
|
||||
\end{example}
|
||||
|
||||
\begin{example}[Standardtopologie]
|
||||
Jeder normierte Vektorraum $(V, \norm{-})$ über $\mathbb{R}$ ist ein metrischer Raum mit der Metrik
|
||||
\[d : V \times V \to \mathbb{R}, \qquad d(x,y) = \norm{x-y}.\]
|
||||
Insbesondere ist der $\mathbb{R}^n$ ein metrischer Raum mit der durch die \emph{$p$-Norm}
|
||||
\[\norm{x}_p = {(\Sigma_{i=1}^n \vert x_i \vert^p)}^{\frac{1}{p}}\]
|
||||
induzierten Metrik.
|
||||
Für $p=2$ erhält man so die \emph{euklidische Metrik} auf dem $\mathbb{R}^n$, die zugehörige Topologie nennt man die \emph{Standardtopologie} $\CO_{std}$ auf $\mathbb{R}^n$.
|
||||
\end{example}
|
||||
|
||||
\begin{lemma}
|
||||
Sei $(X,d)$ ein metrischer Raum.
|
||||
Für jede Teilmenge $M \subseteq X$ mit der Metrik $d\vert_{M \times M}$ stimmt die metrische Topologie mit der Teilraumtopologie überein.
|
||||
Insbesondere erhält man so die \emph{Standardtopologie} auf Teilmengen $M \subseteq \mathbb{R}^n$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}
|
||||
Zwei Metriken $d_1, d_2$ auf einer Menge $X$ induzieren genau dann die gleiche Topologie auf $X$, wenn es zu jedem $x \in X$ und jedem $\epsilon > 0$ ein $\delta > 0$ gibt mit
|
||||
\[d_1(x,y) < \delta \Rightarrow d_2(x,y) < \epsilon \qquad \land \qquad d_2(x,y) < \delta \Rightarrow d_1(x,y) < \epsilon.\]
|
||||
In diesem Fall nennen wir die Metriken \emph{äquivalent}.
|
||||
\end{theorem}
|
||||
|
||||
\subsection{Umgebungen}
|
||||
\begin{definition}[Umgebung]
|
||||
Sei $(X, \CO)$ ein topologischer Raum.
|
||||
Eine Menge $U \subseteq X$ heißt \emph{Umgebung} eines Punktes $x \in X$, wenn eine offene Menge $O \in \CO$ existiert mit $x \in O \subseteq U$.
|
||||
Die Menge der Umgebungen von $x \in X$ nennen wir $\CU(x)$.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
Für jeden topologischen Raum $(X, \CO)$ und $x \in X$ gilt:
|
||||
\begin{enumerate}
|
||||
\item Ist $U \in \CU(x)$ so ist auch jedes $V \subseteq U$ eine Umgebung von $x$.
|
||||
\item Für jede offene Menge $O \in \CO$ gilt $x \in O \Rightarrow O \in \CU(x)$.
|
||||
\item Jede Umgebung von $x$ enthält eine offene Umgebung von $x$.
|
||||
\item Vereinigungen von Umgebungen von $x$ sind Umgebungen von $x$.
|
||||
\item Endliche Schnitte von Umgebungen von $x$ sind Umgebungen von $x$.
|
||||
\item Eine Teilmenge $O \subseteq X$ ist genau dann offen, wenn sie Umgebung aller ihrer Punkte ist.
|
||||
\end{enumerate}
|
||||
\end{remark}
|
||||
|
||||
\subsection{Inneres, Abschluss und Rand}
|
||||
\begin{definition}[Inneres, Abschluss, Rand]
|
||||
Sei $(X, \CO)$ ein topologischer Raum.
|
||||
\begin{itemize}
|
||||
\item Das \emph{Innere} $\inter{M}$ einer Teilmenge $M \subseteq X$ ist:
|
||||
\[\inter{M} = \bigcup_{O\in \CO,O\subseteq M} O.\]
|
||||
\item Der \emph{Abschluss} $\clos{M}$ einer Teilmenge $M \subseteq X$ ist:
|
||||
\[\clos{M} = \bigcap_{A \subseteq X \text{ abg.}, M\subseteq A} A.\]
|
||||
\item Der \emph{Rand} $\bound{M}$ einer Teilmenge $M \subseteq X$ ist:
|
||||
\[\bound{M} = \clos{M} \setminus \inter{M}.\]
|
||||
\end{itemize}
|
||||
Eine Teilmenge $M \subseteq X$ heißt \emph{dicht} in $X$, wenn $\clos{M} = X$ und \emph{nirgend dicht} in $X$, wenn $\inter{\clos{M}} = \emptyset$.
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}
|
||||
Es gelten folgende Rechenregeln:
|
||||
\[\inter{M} = X \setminus \clos{X \setminus M}\]
|
||||
und
|
||||
\[\bound{M} = \clos{M} \cap \clos{X \setminus M} = \bound{X \setminus M}.\]
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}
|
||||
Sei $(X, \CO)$ ein topologischer Raum und $M \subseteq X$ eine Teilmenge. Dann gilt:
|
||||
\begin{enumerate}
|
||||
\item $\inter{M}$ ist offen, $\inter{M} \subseteq M$ und $\inter{M} = M$ genau dann, wenn $M$ offen ist.
|
||||
\item $\clos{M}$ ist abgeschlossen, $M \subseteq \clos{M}$ und $\clos{M} = M$ genau dann, wenn $M$ abgeschlossen ist.
|
||||
\item $\bound{M}$ ist abgeschlossen, $\bound{\bound{M}} \subseteq \bound{M}$ und $\bound{\bound{\bound{M}}} = \bound{\bound{M}}$
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}
|
||||
Sei $(X, \CO)$ ein topologischer Raum und $M \subseteq X$. Es gilt:
|
||||
\begin{enumerate}
|
||||
\item $x \in \inter{M} \iff$ Es gibt eine Umgebung $U \in \CU(x)$ mit $U \subseteq M$. $\iff M$ ist eine Umgebung von $x$.
|
||||
\item $x \in \clos{M} \iff$ Für jede Umgebung $U \in \CU(x)$ ist $U \cap M \not= \emptyset$. $\iff X \setminus M$ ist keine Umgebung von $x$.
|
||||
\item $x \in \bound{M} \iff$ Für alle $U \in \CU(x)$ gilt $U \cap M \not= \emptyset$ und $U \cap (X \setminus M) \not= \emptyset$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\section{Vergleich und Erzeugung von Topologien}
|
||||
\subsection{Vergleich von Topologien}
|
||||
\begin{definition}[Fein- und Grobheit]
|
||||
Sei $X$ eine Menge mit Topologien $\CO_1, \CO_2$. Wenn $\CO_2 \subseteq \CO_1$ gilt nennen wir
|
||||
\begin{itemize}
|
||||
\item $\CO_1$ \emph{feiner} als $\CO_2$
|
||||
\item $\CO_2$ \emph{gröber} als $\CO_1$
|
||||
\end{itemize}
|
||||
\end{definition}
|
||||
|
||||
\begin{example}[Feinste Topologie]
|
||||
Die diskrete Topologie $\CO_{dsk} = \CP(X)$ ist die \emph{feinste} Topologie auf $X$.
|
||||
\end{example}
|
||||
|
||||
\begin{example}[Gröbste Topologie]
|
||||
Die indiskrete Topologie $\CO_{ind} = \{\emptyset, X\}$ ist die \emph{gröbste} Topologie auf $X$.
|
||||
\end{example}
|
||||
|
||||
\begin{example}[Kofinite und Standardtopologie]
|
||||
Die kofinite Topologie $\CO_{kof}$ auf einer Teilmenge $X \subseteq \mathbb{R}^n$ ist gröber als die Standardtopologie $\CO_{std}$.
|
||||
\end{example}
|
||||
|
||||
\begin{theorem}
|
||||
Schnitte von Topologien sind wieder Topologien.
|
||||
Der Schnitt von Topologien $\CO_i$ ist gröber als jedes $\CO_i$.
|
||||
\end{theorem}
|
||||
|
||||
\subsection{(Sub-) Basen}
|
||||
\begin{definition}[Erzeugte Topologie]
|
||||
Für jede Teilmenge $\CM \subseteq \CP(X)$ ist die von $\CM$ \emph{erzeugte Topologie} definiert als
|
||||
\[\langle \CM \rangle = \bigcap_{\CM \subseteq \CO \subseteq \CP(X), \CO \text{ Topologie}}\CO.\]
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
$\langle \CM \rangle$ ist eine Topologie auf $X$ und zwar die gröbste Topologie die $CM$ enthält.
|
||||
\end{theorem}
|
||||
|
||||
\begin{lemma}
|
||||
Für jede Menge $X$ und $\CM \subset \CP(X)$ lässt sich $\langle \CM \rangle$ auch charakterisieren als
|
||||
\[\langle \CM \rangle = \{\text{beliebige Vereinigungen von endlichen Schnitten von Mengen in }\CM\}.\]
|
||||
\end{lemma}
|
||||
|
||||
\begin{definition}[(Sub-) Basis]
|
||||
Sei $(X, \CO)$ ein topologischer Raum. Eine Teilmenge $\CM \subseteq \CP(X)$ heißt
|
||||
\begin{enumerate}
|
||||
\item \emph{Subbasis} von $\CO$, wenn $\CO = \langle \CM \rangle$ gilt.
|
||||
\item \emph{Basis} von $\CO$, wenn $\CM \subseteq \CO$ und jede Menge in $\CO$ eine Vereinigung von Mengen aus $\CM$ ist.
|
||||
\end{enumerate}
|
||||
Besitzt $\CO$ eine abzählbare Basis, so sagt man $\CO$ erfülle das \emph{2. Abzählbarkeitsaxiom}.
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}
|
||||
Eine Teilmenge $\CB \subseteq \CP(X)$ ist Basis einer Topologie auf $X$ genau dann, wenn
|
||||
\begin{enumerate}
|
||||
\item zu jedem $x \in X$ gibt es ein $B \in \CB$ mit $x \in B$,
|
||||
\item für alle $B_1, B_2 \in \CB$ und alle $x \in B_1 \cap B_2$ gibt es ein $B_3 \in \CB$ mit $x \in B_3 \subseteq B_1 \cap B_2$.
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
|
||||
\begin{example}[Basis der (In-) diskreten Topologie]
|
||||
Für jede Menge $X$ ist $\CM = \{ X\}$ eine Basis der indiskreten Topologie und $\CM' = \{\{ x \} \;\vert\; x \in X\}$ eine Basis der diskreten Topologie.
|
||||
\end{example}
|
||||
|
||||
\begin{example}[Basis der Standardtopologie]
|
||||
Die Menge $\CM = \{(a,b) \subseteq \mathbb{R} \;\vert\; a < b \in \mathbb{R}\}$ ist eine Basis der Standardtopologie auf $\mathbb{R}$.
|
||||
Außerdem ist die Menge $\CM = \{(a,b) \subseteq \mathbb{R} \;\vert\; a < b \in \mathbb{Q}\}$ eine Basis der Standradtopologie auf $\mathbb{R}$, somit erfüllt die Standardtopologie das 2. Abzählbarkeitsaxiom.
|
||||
\end{example}
|
||||
|
||||
\begin{lemma}[Einbettungssatz von Urysohn]
|
||||
Erfüllt ein topologischer Raum $(X, \CO)$ das 2. Abzählbarkeitsaxiom, so gibt es eine abzählbare dichte Teilmenge $M \subseteq X$.
|
||||
\end{lemma}
|
||||
|
||||
\section{Stetige Abbildungen}
|
||||
1
tex/sections/02_zusammenhang.tex
Normal file
1
tex/sections/02_zusammenhang.tex
Normal file
|
|
@ -0,0 +1 @@
|
|||
\chapter{Zusammenhang und Trennung}
|
||||
1
tex/sections/03_konstruktion.tex
Normal file
1
tex/sections/03_konstruktion.tex
Normal file
|
|
@ -0,0 +1 @@
|
|||
\chapter{Konstruktion von Topologischen Räumen}
|
||||
1
tex/sections/04_kompaktheit.tex
Normal file
1
tex/sections/04_kompaktheit.tex
Normal file
|
|
@ -0,0 +1 @@
|
|||
\chapter{Kompaktheit}
|
||||
Loading…
Reference in a new issue