Topology

Selected questions

Preliminary Materials

As the study of the properties of spaces that undergo continuous deformations, topology is a major branch of mathematics born from geometry. Examine the foundations of relations, maps, orders, and sets. Explore basic topology on Euclidian space $\mathbb{R}^n$, the set properties of closure and compactness, and how functions are continuous between two topological spaces.

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Topological Spaces

Topological spaces are mathematical spaces defined by a set of specific axioms relating points and neighborhoods. Discover how to define and compare topological spaces and explore the concepts of metric and normed spaces. Follow up with a more in depth examination of both open and closed sets, and the associated properties of closure, interior, boundary, and denseness.

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Convergence and Continuity

Continuity of topological spaces is analogous to classical limits. Explore how sequences in topology display convergence and how this leads to a definition of continuity. Apply this to homeomorphisms, continuous functions between topological spaces whose inverses are also continuous.

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Compact Spaces

Compactness of topological spaces is akin to finiteness of regular functions and gives spaces a quality of being well-behaved. Determine first how to define compact spaces and, more specifically, sequential compactness. Explore subspaces and define relative and local compactness.

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Quotient and Product Spaces

Quotient and product spaces focus on creating new topological spaces from others. Learn how to "glue" points of a topological space together to create quotient spaces and how cartesian products of topological spaces can create product spaces. Apply properties of compactness to quotient and product spaces using Tychonoff's Theorem.

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Separation Axioms

Separation axioms are a series of restrictions that can be places on topological spaces to help classify different spaces. Learn about the $T_n$ axioms from least to most restrictive and how they apply to regular spaces.

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Function Spaces

A function space is a set of functions between two sets. Explore what it means for a sequence of functions to converge to another function and utilize the Arzela­-Ascoli Theorem to determine convergence. Apply compact open topology to function maps between two topological spaces.

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Fundamental Group

The fundamental group is a mathematical group that determines if two paths with shared end points can be continuously deformed into each other, also known as homotopy of functions. Define and apply the fundamental group and discover how homomorphisms are induced.

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Seifert-van Kampen Theorem

The Seifert-van Kampen Theorem allows for the analysis of the fundamental group of spaces that are constructed from simpler ones. Construct new groups from other groups using the free product and apply the Seifert-van Kampen Theorem. Explore basic 2D cell complexes.

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Covering Spaces

Covering maps, as applied to homotopy groups, are continuous functions between two topological spaces in which each point from one "covers" or is "covered" by a point in the other, essentially a local homomorphism. Define and construct covering spaces, classify them, and learn to apply deck transformations.

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