Topology is the study of the alteration of mathematical objects. The field of topology is similar to that of geometry in that objects exist in a spatial dimension and are analyzed from that point of view. Topology, being an advanced topic in mathematics, has a small handful of math prerequisites with which the hopeful student should be familiar.
Topology is a natural extension of geometry. To understand the processes involved in the alterations of topological objects, a student should have a solid understanding of measurements in geometry, including length, area, volume and arc length. Different calculations of distance play an important part in topology, so the student should understand at least some of the basic distance types used in geometry such as Euclidean distance. In addition, topology makes use of the advanced shapes and objects contained in geometry, so a student preparing to study topology should review these objects and their properties.
Set theory is the basic language of topology. When a topologist works, she is usually working with the language of set theory to describe the spatial characteristics of topological objects. Hence, an enduring aptitude with set theory is required throughout the field of topology. Before studying topology, the student should review the axioms of set theory as well as the basic theories and results related to these axioms. Most topology uses the axiom of choice, so students should be familiar with this controversial axiom as well.
Most of the work in a topology class relates to finding and understanding topological proofs. A sound understanding of writing and reading proofs is necessary for success in a topology course. A student preparing to enroll in a topology course should familiarize himself with all of the standard proof techniques in mathematics. The most important proof methods in topology are direct proofs, proof by exhaustion, proof by induction and proof by contradiction.
Algebra is essential for mathematically describing the objects investigated in topology. Not only should a student know the rules of algebraic operations but she should also understand the analytical techniques of algebra, including the reduction of equations and how to classify mathematical objects in algebraic terms. Before enrolling in topology, review basic algebra and its structures such as rings, groups and fields. In addition, although not all courses on topology heavily base their content on abstract algebra, a strong understanding of how abstract algebra can be used to analyze homeomorphisms is particularly useful in topology.
- "Introduction to Topology and Geometry"; Saul Stahl; 2005
- "Introduction to Topology"; V.A. Vassiliev; 2001
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