Introduction to Topology

Code:

M3008

Acronym:

M3008

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	56	162
L:CC	0	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	0	Official Study Plan	2	-	6	56	162
			3
L:G	0	study plan from 2017/18	3	-	6	56	162
L:M	24	Official Study Plan	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

Familiarity with the area of Mathematics which provides the most general and elegant foundations to a fair part of Analysis. Understanding the concept of compactness, one of Topology’s greatest contributions to several other areas of Mathematics.

Learning outcomes and competences

Capacity of solving topology problems of various types. Autonomy in the solution of exercises.

Working method

Presencial

Program

(A) TOPOLOGICAL SPACES

Topological space, open set, closed set, examples, euclidian topology, basis and sub-basis of a topology, limit point, closure, interior, exterior, boundary, dense subset, neighbourhood, connected space, subspace topology, homeomorphism, topological properties, continuous function, continuos image of a connected space, path-connected space, mean value theorem, Brouwer’s fixed point theorem.

(B) METRIC SPACES

Metric space, open and closed balls, topology defined by a metric space, metrizable space, Hausdorff space, convergent and Cauchy sequence, complete metric space, Bolzano-Weierstrass theorem, isometry, isometric embeddding, completion of a metric space, Banach space, Banach’s fixed point theorem.

(C) COMPACTNESS

Covering, compact space, properties of compact spaces, Heine-Borel therorem, every sequence in a compact space admits a convergent subsequence, every compact metric space is complete, uniform continuity, Heine-Cantor theorem.

(D) PRODUCTS AND QUOTIENTS

Product topology, open and closed functions, projections and their properties, Tikhonov’s theorem (finite case), Heine-Borel theorem for higher dimensions, quotient function and quotient space, theorems involving compactness.

Mandatory literature

Sidney A. Morris; Topology without tears, 2018 (Online, http://www.topologywithouttears.net/topbook.pdf)

Complementary Bibliography

Lima Elon Lages; Espacos métricos
Lima Elon Lages; Elementos de topologia geral
Munkres James R.; Topology. ISBN: 978-1-292-02362-5

Teaching methods and learning activities

Expositional classes with discussion of examples and resolution of exercises.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	52,00
Total:	158,00

Eligibility for exams

No requisites.

Calculation formula of final grade

The syllabus will be divided into two parts. each one evaluated by a test worth 10 points.

The second test is held simultaneously with the first season exam. In the same occasion, it is possible to repeat the first test, and the marks obrtained prevail for those students wishing it.

First season exam:

1. The final mark is the sum of the marks obtained in each test, except possibly in the following case:

2. Marks above 18 require an extra proof (oral or written).

Second season exam:

1. In the second season exam, students may repeat both tests or just one of them (except when they are just trying to improve their mark).

2. The mark of each part is the maximum of the marks obtained in the respective tests in both seasons (except when they are just trying to improve their mark).

3. The final mark of the second season is the sum of the marks obtained in both parts, rounded to integers, except possibly in the following cases:

4. Students having obtained a mark equal or above 8,0 and below 9,5 have access to a complementary proof to decide if they are approved (with 10 points) or if they fail (with 8 or 9 points).

5. Marks above 18 require an extra proof (oral or written).