Website owner:  James Miller

Topology

Concepts of Topology. Topological property. Topological transformation. Continuous deformation. Genus. Homeomorphism.

Euler's formula. Euler characteristic. Triangulation of a surface. Compatible triangle orientations. Orientable surfaces.

Set theory. Union, intersection, complement, difference. Venn diagram. Algebra of sets. Countable set. Cardinality. Cartesian product. Partition. Partially, linearly and well ordered sets.

Point sets in one, two and three dimensional space. Types of intervals. Open, closed sets. Continuous mappings.

Point sets in one, two, three and n-dimensional Euclidean spaces. Intervals, neighborhoods, closed sets, open sets, limit points, isolated points. Interior, exterior and boundary points. Derived set. Closure of a set. Perfect set. Arcwise connected sets. Regions. Coverings. Theorems. Bounded, compact sets.

Continuous functions. Sequences. Accumulation point. Limit superior and inferior. Cauchy sequence. Monotonic sequences. Nested intervals. Cantor's principle. Metric space. Uniform convergence of sequences of functions. Theorems.

Set functions

Metric spaces. Examples. Convergence of sequences. Cauchy’s condition for convergence. Complete metric space. Cantor’s Intersection Theorem. Dense sets. Continuous mappings.

Topological spaces

Topological space. Topology. Open and closed sets. Neighborhood. Interior, exterior, limit, boundary, isolated point. Dense, nowhere dense set.

Bases, subbases for a topology. Subspaces. Relative topologies.

Continuous functions in topological spaces. Arbitrary closeness. Sequential continuity at a point. Open and closed mappings. Homeomorphism. Topological transformation.

Metric spaces as topological spaces. Equivalent metrics. Metrization problem. Isometric metric spaces.

First and Second Countable spaces. Cover. Lindelof space. Dense set. Separable space. Hereditary property.

Separation axioms. T1-Space. Cofinite topology. Hausdorff space. Regular and normal spaces. Urysohn's Lemma and Metrization Theorem. Completely regular space. Tychonoff space.

Compactness. Cover. Heine-Borel Theorem. Finite intersection property. Sequentially, countably and locally compact spaces. Bolzano-Weierstrass Theorem. Compactum. Compactification.

Cartesian product. Projection function. Product topology. Product space. Subbase and base for product topology. Metric product spaces

Connectedness. Connected, locally connected and disconnected sets and spaces. Components. Homotopic paths. Theorems.