Lectures related to Interior and Closure in Topology

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Numerical Analysis and Optimization: Concepts of Distance and Subsets

Introduces key concepts in numerical analysis and optimization, focusing on distances, subsets, and their properties in R^n.

Dimension Theory: Topological Space and Closed Subsets

Explores dimension theory in topological spaces and closed subsets, including height interpretation and additivity.

Point Interior and Boundary in Real Numbers

Explores point interior and boundary concepts in real numbers, covering definitions and implications.

Interior Points and Compact Sets

Explores interior points, boundaries, adherence, and compact sets, including definitions and examples.

Metric Spaces: Topology and Continuity

Introduces metric spaces, topology, and continuity, emphasizing the importance of open sets and the Hausdorff property.

Functional Analysis I: Foundations and Applications

Covers the foundations of modern analysis, introductory functional analysis, and applications in MAB111.

Lifting Criterion

Explores the lifting criterion for maps between path-connected and locally path-connected spaces.

Base B for X: Connected Sets and Open Bases

Explores the concept of a base B for a topological space X and the properties of connected sets and open bases.

Topology: Open Sets, Compactness, and Connectivity

Explores open sets, compactness, and connectivity in topology, covering topological spaces and compact sets.

Sub-Varieties and Topological Concepts

Covers sub-varieties in different dimensions and topological concepts with examples and applications of theorems.

Homotopy Extension Property

Introduces the homotopy extension property, exploring conditions for extending continuous maps.

Modular curves: Riemann surfaces and transition maps

Covers modular curves as compact Riemann surfaces, explaining their topology, construction of holomorphic charts, and properties.

CW Complexes

Covers the construction and properties of CW complexes, including weak topology and characteristic maps.

Dimension theory of rings

Covers the dimension theory of rings, including additivity of dimension and height, Krull's Hauptidealsatz, and the height of general complete intersections.

Sheaves: Hartshorne I.1

Covers the concept of sheaves, emphasizing the unique determination of functions by local data and the importance of direct limits.

Homology of Riemann Surfaces

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Advanced Analysis 2: Isolated Points and Continuity

Covers advanced topics in analysis, focusing on isolated points, continuity, and level sets.

Closed Curves and Topological Spaces

Explores closed curves in topological spaces, emphasizing their properties and significance in mathematics.

Topology of Riemann Surfaces

Covers the topology of Riemann surfaces, focusing on orientation and orientability.