Point-Set Topology

These are the notes about point-set topology from Munkres.

Set Theory

Sets

• Sets: Elements, objects, Subsets, proper subsets, Union, intersection, disjoint, empty set.
• Contrapositive, Converse and Negation
• Complement of a set relative to other set: $A-B$
• DeMorgan’s Law: $A-(B\cup C),A-(B\cap C)$
• For a collection of set S, Arbitrary Union ${\bigcup }_{s\in S}s$ and Intersection ${\bigcap }_{s\in S}s$
• Cartesian Product of Sets: $A\times B$
  Functions
• Def: A function: $f:X\to Y$ is a subset of $X\times Y$ with each $x\in X$ appearing exactly once as the first coordinate of an ordered pair in this subset.
  • Ex: $f:\mathbb{R}\to \mathbb{R}$ defined as $x\mapsto x^2+1$. Then the subset $(x,x^2+1)\in \mathbb{R}\times \mathbb{R}$. A different function $f_2:\mathbb{R}\to {\mathbb{R}}_{+},x\mapsto x^2+1$
• Restriction of $f:A\to B$ to subset $A^{\prime }$ is defined as $f{|}_{A^{\prime }}:A^{\prime }\to B$.
• Composite Function $f\circ g$
• Injectivity, Surjectivity and Bijectivity
• Image, Pre-image of a set
• Example: $f(x)=3x^2+1$.
  Relation
• Def: A relation on a set A is a subset $C\subset A\times A$
• Def: Equivalence relation $\sim$ has three properties:
  • Reflexivity, Symmetry, Transitivity
• Def: Equivalence class, subset $E=\{y|y\sim x\}$
  • Equivalence classes of set A form a partition (disjoint nonempty subsets whose union is A).
  • Given partition D of A, there is exactly one equivalence relation on A from which it is derived.
    • Prove existence and uniqueness of equivalence relation from partition D.
• Def: Order Relation $\le \subset A\times A$ is a relation satisfying
  • Comparability: if $x\ne y$, Either $x\le y$ or $y\le x$
  • Non-reflexivity: $x\le y$ and $y\le x$ implies $x=y$
  • Transitivity: $x\le y\le z⟹x\le z$
• Def: Order Type
• Def: Dictionary order relation
  Integers and $\mathbb{R}$
• Def: Field, ordered field, linear continuum
• Least upper bound property and archimedes principle is one of the most important properties of $\mathbb{R}$
• Inductive set: If $x\in \mathbb{R}$ and $1\in \mathbb{R}$, then $x+1\in \mathbb{R}$, where $+$ is the binary operation for the field.

Resources

• Munkres, Topology (2nd Edition)
• Counterexamples of Topology
• John M. Lee, Introduction to Topological Manifolds (2nd Edition)
• MTG 4302/5316, Introduction to Topology I, Fall 2023 – Henry Adams
• ICTP - Bruno Zimmerman