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Programme & Unit Catalogues

## MA50055: Introduction to topology

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Masters UG & PG (FHEQ level 7)
Period: Semester 2
Assessment Summary: CW 25%, EX 75%
Assessment Detail:
• Coursework (CW 25%)
• Examination (EX 75%)
Supplementary Assessment: Mandatory extra work (where allowed by programme regulations)
Requisites: Before taking this module you must take MA30041 or equivalent.
Description: Aims:
This course provides an introduction to the foundations of topology, culminating with a sketch of the classification of compact surfaces. As such it provides a self-contained account of one of the triumphs of 20th century mathematics.

Learning Outcomes:
Students successfully completing this course will be able to:
* state definitions and prove fundamental theorems about topological spaces and related concepts.
* relate and distinguish between topological and metric spaces, with an understanding of their roles in analysis.
* apply their understanding of connected, compact and Hausdorff spaces to unfamiliar problems.
* classify compact surfaces and recognise surfaces in the classification theorem. Through their coursework, students will be able to demonstrate an in depth understanding of a major topic in topology and an appreciation of the role of topology in mathematics.

Skills:
Analytic skills T/F A; Problem solving T/F A; Written communication F A.

Content:
1. Core topics: topologies and topological spaces, subspaces, continuous maps and homeomorphisms, connected spaces, compact spaces, Hausdorff spaces.
2. Topics from the following: initial and final topologies, bases and sub-bases, product spaces, disjoint unions, quotient spaces, equalisers and coequalisers, pullbacks and pushouts, function spaces (compact-open topology), characterisations of compactness, axiom of choice and Zorn's lemma, Tychonoff's theorem, path connectedness, local separation and connectedness properties.
3. Compact surfaces and their representation as quotient spaces. Sketch of the classification of compact surfaces.
4. An in depth study of a major topic of topology, such as compactness.
Programme availability:

#### MA50055 is Optional on the following programmes:

Department of Mathematical Sciences

Notes:
* This unit catalogue is applicable for the 2015/16 academic year only. Students continuing their studies into 2016/17 and beyond should not assume that this unit will be available in future years in the format displayed here for 2015/16.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.