Introduction to Topology

Code:

M353

Acronym:

M353

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2014/2015 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/search.php?search=Introdu%C3%A7%C3%A3o+%C3%A0+Topologia
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:M	6	Plano de estudos a partir de 2009	3	-	7,5	-

Teaching language

Portuguese

Objectives

Deepen and unify knowledge obtained in different disciplines of calculus and generalize some of the concepts and results into a context that we may consider "essential." More precisely,  come to know and apply in the resolution of problems some core concepts of topology that come as generalizations from the calculus, Euclidean geometry and linear algebra and analytic geometry, namely about continuity of functions, convergence and limits, connectivity,
compactness and completeness.

Learning outcomes and competences

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Working method

Presencial

Program

Metric spaces. Continuity and convergence. Isometries and homeomorphisms. Metric
and topological concepts: open and closed sets; neighborhoods; closure and interior;
boundary.Topologies, bases and sub-bases. Products of topological spaces.

Connected spaces; connected components; products of connected spaces;
connectivity and topological invariants; pathwise connected sets; continuous functions
in connected spaces and generalizations of the intermediate value theorem of
Bolzano.

Limits of sequences. Sequences of real numbers. Convergence and topology. Sequences of functions. Cartesian infinite products. Limits of functions. Complete Metric Spaces. Cauchy sequences. Complete metric spaces. Compact: definition and consequences.

Mandatory literature

Lima Elon Lages; Espacos metricos (The recommended book will be followed throughout the course)

Comments from the literature

The recommended book will be followed in general throughout the course

Teaching methods and learning activities

The contact hours consist of theoretical and practical lessons, allowing teachers to
organize and manage the time available for presenting the subject matter, solving
problems, accomplishing tasks and practical work.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	60,00
Frequência das aulas	60,00
Total:	120,00

Calculation formula of final grade

Examination, where two groups of  questions, each one graded 4/20, may be replaced, by the student's choice, by the results of two mini-exams, except in the case of exams made for the purpose of obtaining better marks or, in special occasions, for the conclusion of the course. The other special exams migth be either oral or written.

 

Special assessment (TE, DA, ...)

Written or oral examination.

Classification improvement

Examination.