Code: | M3008 | Acronym: | M3008 | Level: | 300 |
Keywords | |
---|---|
Classification | Keyword |
OFICIAL | Mathematics |
Active? | Yes |
Responsible unit: | Department of Mathematics |
Course/CS Responsible: | Bachelor in Mathematics |
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 |
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.
Capacity of solving topology problems of various types. Autonomy in the solution of exercises.
(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.
designation | Weight (%) |
---|---|
Teste | 100,00 |
Total: | 100,00 |
designation | Time (hours) |
---|---|
Estudo autónomo | 106,00 |
Frequência das aulas | 52,00 |
Total: | 158,00 |