\contentsline {part}{I\hspace {1em}Fundamentals}{1}{part.1}%
\contentsline {chapter}{\numberline {1}Topological spaces}{3}{chapter.1}%
\contentsline {section}{\numberline {1.1}Topology}{3}{section.1.1}%
\contentsline {subsection}{\numberline {1.1.1}Comparing topologies}{5}{subsection.1.1.1}%
\contentsline {subsection}{\numberline {1.1.2}Examples of topological spaces}{6}{subsection.1.1.2}%
\contentsline {subsection}{\numberline {1.1.3}Basis of a topological space}{7}{subsection.1.1.3}%
\contentsline {subsection}{\numberline {1.1.4}Euclidean topology}{8}{subsection.1.1.4}%
\contentsline {chapter}{\numberline {2}Limit points}{11}{chapter.2}%
\contentsline {section}{\numberline {2.1}Neighborhoods}{11}{section.2.1}%
\contentsline {section}{\numberline {2.2}Limit points}{12}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}Constructions from limit points}{12}{subsection.2.2.1}%
\contentsline {section}{\numberline {2.3}Properties of closure}{13}{section.2.3}%
\contentsline {section}{\numberline {2.4}Properties of interior}{14}{section.2.4}%
\contentsline {section}{\numberline {2.5}Equivalent definitions}{14}{section.2.5}%
\contentsline {chapter}{\numberline {3}Continuous functions}{17}{chapter.3}%
\contentsline {section}{\numberline {3.1}Continuous function (Topological space)}{17}{section.3.1}%
\contentsline {section}{\numberline {3.2}Homeomorphisms}{18}{section.3.2}%
\contentsline {section}{\numberline {3.3}Embeddings}{19}{section.3.3}%
\contentsline {chapter}{\numberline {4}Constructions of topological spaces}{21}{chapter.4}%
\contentsline {section}{\numberline {4.1}Topological subspaces}{21}{section.4.1}%
\contentsline {section}{\numberline {4.2}Product topological spaces}{22}{section.4.2}%
\contentsline {section}{\numberline {4.3}Quotient topological spaces}{23}{section.4.3}%
\contentsline {section}{\numberline {4.4}Final topology}{23}{section.4.4}%
\contentsline {chapter}{\numberline {5}Metric spaces}{25}{chapter.5}%
\contentsline {section}{\numberline {5.1}Metric space}{25}{section.5.1}%
\contentsline {section}{\numberline {5.2}Metric space}{25}{section.5.2}%
\contentsline {section}{\numberline {5.3}Topological space induced by a metric space}{26}{section.5.3}%
\contentsline {subsection}{\numberline {5.3.1}Open balls}{27}{subsection.5.3.1}%
\contentsline {section}{\numberline {5.4}Continuous function (Metric space)}{28}{section.5.4}%
\contentsline {section}{\numberline {5.5}Complete metric space}{29}{section.5.5}%
\contentsline {section}{\numberline {5.6}Isometries}{30}{section.5.6}%
\contentsline {section}{\numberline {5.7}Banach fixed-point theorem}{30}{section.5.7}%
\contentsline {section}{\numberline {5.8}Baire spaces}{31}{section.5.8}%
\contentsline {section}{\numberline {5.9}Metrizations}{31}{section.5.9}%
\contentsline {section}{\numberline {5.10}Open and closed sets}{31}{section.5.10}%
\contentsline {section}{\numberline {5.11}Origins of topologies}{33}{section.5.11}%
\contentsline {chapter}{\numberline {6}Connected spaces}{35}{chapter.6}%
\contentsline {section}{\numberline {6.1}Connected space}{35}{section.6.1}%
\contentsline {subsection}{\numberline {6.1.1}Connected sets}{36}{subsection.6.1.1}%
\contentsline {section}{\numberline {6.2}Path connected space}{37}{section.6.2}%
\contentsline {section}{\numberline {6.3}Locally connected spaces}{37}{section.6.3}%
\contentsline {chapter}{\numberline {7}Compact spaces}{39}{chapter.7}%
\contentsline {section}{\numberline {7.1}Compact space}{39}{section.7.1}%
\contentsline {section}{\numberline {7.2}Sequentially Compact space}{40}{section.7.2}%
\contentsline {section}{\numberline {7.3}Other?}{41}{section.7.3}%
\contentsline {section}{\numberline {7.4}Compactifications}{41}{section.7.4}%
\contentsline {chapter}{\numberline {8}Separation properties}{43}{chapter.8}%
\contentsline {section}{\numberline {8.1}Kolmogorov space}{43}{section.8.1}%
\contentsline {section}{\numberline {8.2}Fréchet spaces}{43}{section.8.2}%
\contentsline {section}{\numberline {8.3}Hausdorff spaces}{44}{section.8.3}%
\contentsline {section}{\numberline {8.4}Regular spaces}{44}{section.8.4}%
\contentsline {chapter}{\numberline {9}Countability properties}{45}{chapter.9}%
\contentsline {section}{\numberline {9.1}First countable spaces}{45}{section.9.1}%
\contentsline {section}{\numberline {9.2}Second countable spaces}{45}{section.9.2}%
\contentsline {chapter}{\numberline {10}Topological sequences}{47}{chapter.10}%
\contentsline {section}{\numberline {10.1}Topological sequences}{47}{section.10.1}%
\contentsline {section}{\numberline {10.2}Nets}{47}{section.10.2}%
\contentsline {section}{\numberline {10.3}Filters}{47}{section.10.3}%