M.Sc. Mathematics - Topology Exam Prep

Definitions And Classification

• Topic: The Three Axioms of Topology

  Details: A collection $\tau \subseteq \mathcal{P}(X)$ must satisfy: 1. $\emptyset, X \in \tau$; 2. If $\{U_i\}_{i \in I} \subseteq \tau$, then $\cup_{i \in I} U_i \in \tau$ (Closed under arbitrary unions); 3. If $U, V \in \tau$, then $U \cap V \in \tau$ (Closed under finite intersections).

  Pro Tip: If a question asks if a collection is a topology, check the union first. Often, the union of two sets in the collection will fail to be present.
• Topic: Special Topologies

  Types

  • Name: Cofinite Topology ($\tau_{cf}$)

    Rule: $U$ is open iff $X \setminus U$ is finite or $U = \emptyset$.

    Fact: If $X$ is finite, the cofinite topology is the same as the Discrete topology.
  • Name: Cocountable Topology ($\tau_{cc}$)

    Rule: $U$ is open iff $X \setminus U$ is countable or $U = \emptyset$.

    Fact: On an uncountable set, singletons are NOT open, but they are closed.
  • Name: Sorgenfrey Line (Lower Limit Topology)

    Rule: Generated by the basis $[a, b)$.

    Fact: This is strictly finer than the standard topology on $\mathbb{R}$.

Topology Set Operators

• Operator: Closure ($\overline{A}$)

  Properties

  • $\overline{A} = A \cup A'$ (where $A'$ is the set of limit points).
  • $\overline{A}$ is the smallest closed set containing $A$.
  • $\overline{A \cup B} = \overline{A} \cup \overline{B}$.
  • $\overline{A \cap B} \subseteq \overline{A} \cap \overline{B}$ (Note: Equality may not hold).
• Operator: Interior ($Int(A)$)

  Properties

  • $Int(A)$ is the largest open set contained in $A$.
  • $Int(A) = X \setminus \overline{(X \setminus A)}$.
  • $Int(A \cap B) = Int(A) \cap Int(B)$.
  • $Int(A \cup B) \supseteq Int(A) \cup Int(B)$ (Note: Equality may not hold).
• Operator: Boundary ($Bd(A)$)

  Logic: Points $x$ such that every neighborhood of $x$ hits both $A$ and $X \setminus A$.

  Theorem: $Bd(A) = \emptyset$ if and only if $A$ is both open and closed (clopen).

Fundamental Theorems

• Name: Kuratowski Closure Axioms

  Description: A set $X$ can be given a topology by defining a closure operator $c: \mathcal{P}(X) \to \mathcal{P}(X)$ satisfying: 1. $c(\emptyset) = \emptyset$; 2. $A \subseteq c(A)$; 3. $c(c(A)) = c(A)$; 4. $c(A \cup B) = c(A) \cup c(B)$.
• Name: Basis Criteria

  Theorem: A collection $\mathcal{B}$ of subsets of $X$ is a basis for a topology iff: 1. $\cup_{B \in \mathcal{B}} B = X$; 2. If $x \in B_1 \cap B_2$, there exists $B_3 \in \mathcal{B}$ such that $x \in B_3 \subseteq B_1 \cap B_2$.
• Name: Continuity Theorem

  Characterizations

  • $f$ is continuous iff $f(\overline{A}) \subseteq \overline{f(A)}$ for all $A \subseteq X$.
  • $f$ is continuous iff $f^{-1}(Int(B)) \subseteq Int(f^{-1}(B))$ for all $B \subseteq Y$.
  • $f$ is continuous iff preimage of every basis element is open.

Exam Tricks And Shortcuts

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