Introduction to Topology

Code:

M3008

Acronym:

M3008

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2016/2017 - 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:M	10	Official Study Plan	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Portuguese

Objectives

To introduce the theory of metric and topological spaces; to show that metric spaces are more general than Euclidean spaces and that topological spaces are yet more general than metric spaces.
To introduce some topological invariants to be able to distinguish spaces.

Learning outcomes and competences

To be able to understand that the concepts introduced in the first and second year analysis  can be extended to more general spaces to prove some powerful results used in may parts of mathematics . To be able to understand the concept of topological invariant and its use to distinguish and classify spaces.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Real Analysis I,II and III

Program

Topological spaces: the topology on a space. Examples of topological spaces. The metric spaces. Comparison of topologies. Comparison of metrics. Equivalent metrics. Continuous functions. Continuity and convergence in metric spaces. Homeomorfismos. Supremum of a family of topologies. The topology generated by a collection of parts of a set X. Numerabilidade axioms.

Construction of topological spaces: Initial Topology. Subspaces.Embeddings. Initial topology for a family of maps. Product topology (Tychonoff) and the box topology. Characterization of the product topology. Final topology. Quotient spaces/identification Topology. Examples of quotient spaces. Final topology for a set of maps. Sum topology. Unions of locally finite families of closed sets.

Topological Properties, Invariants: Connected spaces. Path connecteness. Locally connected spaces and locally path connected spaces. Separation axioms. Spaces T0, T1, T2, regular, completely regular and normal. Compact spaces. Compacts in Euclidean Spaces. Sequentially compact spaces. Uniform continuity. Compact and normality. Tychonoff's theorem (demonstration in the finite case). Lebesgue number . Lebesgue's lemma. Locally compact spaces and regularity. Product of metric spaces. Compactifications. Complete metric spaces. Product of complete spaces; complete spaces and compactness; Banach's fixed point theorem. Contraction mappings. Baire’s theorem.Applications of Banach's fixed point theorem.

Mandatory literature

Sutherland W. A.; Introduction to metric and topological spaces. ISBN: 0-19-853161-3
Mendelson Bert; Introduction to topology
Lima Elon Lages 1929-; Espaços métricos. ISBN: 9788524401589
Lima Elon Lages; Elementos de topologia geral
Schubert Horst; Topology. ISBN: 356-02077-0

Comments from the literature

The most important "bibliography" is what is given in the lectures.

Teaching methods and learning activities

Exposure of the program and resolution of exercises.

Resolution, by the students, of the proposed exercises and answering questions about the resolution of problems and proposed work.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Eligibility for exams

Terms of frequency: If the limit of absences is exceeded the student will not be admited access to examination, either in time or normal use (except for students exempted from frequency).

Calculation formula of final grade

Terms of frequency: If the limit of absences is exceeded the student will not be admited access to examination, either in time or normal use (except for students exempted from frequency)

Formula Evaluation: There wil be two components of assessment:

• Continuous Evaluation (optional): based on test results and itcan be corrected by the assessment practices in the classroom (including level of participation and performance in class) *.
• Final written exam
-.-.-.-.-.-.-.-.-.-.-.-.-.-

The evaluation will be done through two tests required and the final exam. Admission to the second test will be conditional upon a minimum grade of 8.0 values.
Minimum grade of second test 6 values
The tests may replace the exam.
The notice of exemption will not necessarily be the arithmetic mean of test scores *
The student with a grade exceeding eighteen values in tests or final examination may eventually be subjected to an extra proof.

For students under normal conditions with access to examination and which have succeeded the distributed evaluation( score above or equal to 10/20), the final classification is obtained by the highest ranking achieved in the distributed evaluation and / or examination

 

Special assessment (TE, DA, ...)

According to the General Evaluation Rules.

Classification improvement

According to the General Evaluation Rules.