# MTH251: Set Topology

Topology is an important branch of mathematics that studies all the “qualitative” or “discrete” properties of continuous objects such as manifolds, i.e. all the properties that aren't changed by any continuous transformations except for the singular (infinitely extreme) ones.

In this sense, topology is a vital arbiter in the “discrete vs continuous” wars. The very existence of topology as a discipline shows that “discrete properties” always exist even if you only work with continuous objects. On the other hand, topology always assumes that these features are “derived” – they're some of the properties of objects and these objects are deeper and that may have many other, continuous properties, too. The topological, discrete properties of these objects are just projections or caricatures of the “whole truth”. (continue reading ...)

At the end of this course the students will be able to understand the theory of metric spaces and topological spaces. They are expected to learn how to write, in logical manner, proofs using important theorems and properties of metric spaces and topological spaces. Students learn to solve problems using the concepts of topology. They present their solutions as rigorous proofs written in correct mathematical English. Students will be able to devise, organize and present brief solutions based on definitions and theorems of topology. Students are expected not only to grasp the concepts of topology and apply them, but also to continue with their overall mathematical development. They will be improving such skills as mathematical writing and the presentation of rigorous logical arguments.

Preliminaries, Metric spaces: Open and closed sets, convergence, completeness.

Continuous and uniformly continuous mappings. Pseudometrics. Fixed point theorem for metric spaces; Topological Spaces. Open bases and sub-bases. Relative topology, Neighborhood system, Limit points, First and second countable spaces. Separable spaces. Products of spaces, Interior, Exterior, Closure and Frontier in product spaces.

Open and closed maps, Continuity and Homeomorphisms, Quotient spaces; Housdorff spaces, regular, and normal spaces, Urysohn's Lemma; Compact spaces, Tychonoff's theorem and locall compact spaces, Compactness for Metric spaces; Connected spaces, Components of a space, Totally disconnected spaces, Local connectedness, Path-wise connectedness

1. James Munkres, Topology (2nd Edition), Prentice Hall, 2000.
1. Sheldon Davis, Topology, McGraw-Hill Science/Engineering/Math, 2004.
2. Seymour Lipschutz, Schaum's Outline of General Topology, McGraw-Hill, 2011.
3. G.F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill, 2004. (link)
4. Stephen Willard, General Topology, Dover Publications, 2004. (link)
5. M.A. Armstrong, Basic Topology, Springer, 2010.