Topology

Topology explores spaces and the study of the continuity of functions, and from the perspective of undergraduates, it can be seen as a generalization of Analysis. Its applications throughout mathematics are nearly as ubiquitous as Set Theory.

General Topology

Topology

• What is a Topological Space? $\left( X, \mathscr{T} \right)$
• Trivial Topology and Discrete Topology
• Subspace Topology / Relative Topology
  • Various Properties of Interiors in Subspaces
• Accumulation Points, Convergence, Complement $a’$
• Various Equivalent Conditions of Interiors $A^{\circ}$
• Separability and Closure $\overline{A}$
• In General Topological Spaces, Limits of Sequences Are Not Unique
  • Coarser Topology and Countable Complement Topology
• Open and Closed Functions

Basis

• Basis and Local Basis $\mathscr{B}$
  • Equivalent Conditions for Bases
  • Topology Generated by a Basis
• Subbasis $\mathscr{S}$
• First Countable and Second Countable
  • First and Second Countability of Metric Spaces
• Proof of Alexander Subbasis Theorem

Topological Properties

Homeomorphism

• What is Continuity?
  • Gluing Lemma
• What is a Homeomorphism?
  • Homeomorphic Mappings Preserve Bases
• Embedding

Topological Properties

• Topological Properties
  • Hereditary Properties
• Fixed Point Properties
• Separation Axioms $T_n$
  • Being a $T1$-Space is Equivalent to All Finite Subsets Being Closed
  • In Hausdorff Spaces, Limits of Sequences are Unique
  • Urysohn’s Lemma and Tietze Extension Theorem

Connectedness

• What is Connectedness?
  • Various Equivalent Conditions of Connected Spaces
  • Surjective Continuous Functions Preserve Connectedness
  • Properties of Connected Subspaces
• Intermediate Value Theorem
• Connected Components and Totally Disconnected Spaces
• What is Path Connectedness?
• Path Connected Components
  • Topologist’s Sine Curve and Comb Space
• Local Connectedness and Local Path Connectedness

Compactness

Many topics in topology are important, but compactness is particularly significant. Even if topology isn’t your strong suit, it’s crucial to diligently study compactness.

• What are Compact and Precompact?
  • Totally Bounded Spaces
  • In Metric Spaces, Being Compact is Equivalent to Being Complete and Totally Bounded
• Limit Point Compactness and Bolzano-Weierstrass Property
• Countable Compactness and Lindelöf Property
• Sequential Compactness
• Finite Intersection Property f.i.p.
• Compact Hausdorff Spaces are Normal Spaces
• Useful Properties of Compact Spaces and Continuous Functions
• Extreme Value Theorem
• Lebesgue’s Number Lemma
• Uniform Continuity Theorem
• Peano Space Filling Theorem
• One Point Compactification

Spaces

Product and Quotient

• Weak Topology
• Quotient Space $X/\sim$
• Product Space $\prod$
• Proof of Tychonoff’s Theorem
• Function Spaces $Y^X$

Manifolds

• What is a Manifold?
• What is a Coordinate System?
• Disjoint Union Topological Spaces

Named Spaces

• Cantor Set
  • Proof of Baire’s Category Theorem
• Torus $T^2 = S^1 \times S^1$

References

• Croom. (1989). Principles of Topology
• Munkres. (2000). Topology(2nd Edition)