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
- 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
- Open and Closed Functions
Basis
- Basis and Local Basis $\mathscr{B}$
- Subbasis $\mathscr{S}$
- First Countable and Second Countable
- Proof of Alexander Subbasis Theorem
Topological Properties
Homeomorphism
Topological Properties
Connectedness
- What is Connectedness?
- Intermediate Value Theorem
- Connected Components and Totally Disconnected Spaces
- What is Path Connectedness?
- Path Connected Components
- 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?
- 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
Named Spaces
References
- Croom. (1989). Principles of Topology
- Munkres. (2000). Topology(2nd Edition)
All posts
- What is a Topological Space?
- Limit Points and Convergence in Topological Spaces, Image Sets
- Self-Evident Topology and Discrete Topology
- Topological Spaces: Separability and Closure
- Limits of sequence are not unique in general Space
- Loose Topology and Leisurely Mountain Topology
- Bases and Local Bases in Topology
- The First Countable and the Second Countable
- First Countability and Second Countability of Metric Spaces
- Subbasis in Topology
- Equivalence Conditions for Bases in Topology
- Continuous in Topology
- Open Functions and Closed Functions
- Homotopy in Topological Spaces
- Topological Properties
- What are Successive Properties in Topology?
- Separation Properties in Topology
- Being a Space and Having All Finite Subsets Closed are Equivalent
- In Hausdorff Spaces, Limits of Sequences Are Unique
- Connectivity in Topology
- Various Equivalent Conditions of Connected Spaces
- The Transfer Continuous Function Preserves Connectivity
- Properties of Subspaces in Connected Spaces
- Connected Components and Totally Disconnected Spaces
- Proof of the Intermediate Value Theorem
- What is the Fixed Point Property in Topology?
- Path-Connectedness in Topology
- Proof of the Adhesive Lemma
- Path Connectivity Components
- Local Connectivity and Local Path Connectivity
- Topologist's Sine Curves and Metric Spaces
- What are Compact and Precompact in Topological Spaces?
- Finite Intersection Property
- A Compact Hausdorff Space is a Normal Space
- Useful Properties of Compact Spaces and Continuous Functions
- Proof of the Maximum and Minimum Value Theorem in Topological Spaces
- Uniform Continuity Theorem
- Additive Compactness and Lindelöf Spaces
- Bolzano-Weierstrass Property and Compactness of Accumulation Points
- Löb's Theorem Proof
- Proof of the Peano Space-Filling Theorem
- Point Compactification
- The Proof of Baire's Category Theorem
- Cantor Set
- Cartesian Product of Topological Spaces
- Proof of the Alexander Subbase Theorem
- Tikhonov's Theorem Proof
- Functional Spaces in Topology
- Wirtinger Inequality and Tietze Extension Theorem
- Quotient Space
- What is a Manifold?
- Separated Union Topological Space
- In Topology, What is a Coordinate System?
- Homomorphism Preserves Basis
- Several Equivalent Conditions for the Interior in a Topological Space
- Subspace Topology, Relative Topology
- Completely Bounded Space
- Generating Topology from a Basis
- Properties of the Interior in Topological Spaces and Subspaces
- The Equivalence between Compact Metric Spaces and Complete, Totally Bounded Spaces
- Embedding in Topology
- Definition of Weak Topology
- What is a Torus in Mathematics?
- What is Sequential Compactness in Topological Spaces?