Subject: Topology

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

1 - Stimulate and develop thinking skills, rigor, deduction and abstraction.
2 - Introduce the study of General Topology, so that the student can generalize the abstract spaces the usual topological concepts (continuity, sequences, convergence, compactness, connectedness, etc.) and the fundamental theorems introduced in previous courses in finite dimensional spaces.
3 - To familiarize students with the evidence of results and require the knowledge of them.

# Syllabus:

1 - Topological spaces
1.1 - Topology on a set. Definition of topological space
1.2 - Open and closed sets
1.3 - Hausdorff topological space
1.4 - Comparable topologies
1.5 - Interior, exterior, boundary, closure and derived from a set
1.6 - Dense set
1.7 - Separable topological space
1.8 - Neighborhood of a point and neighborhood of a set
1.9 - Fundamental system of neighborhoods
1.10 - Regular topological space
1.11 - Basis for a topology
1.12 - Axioms of numerability
2 - Continuity and Limits
2.1 - Continuous functions: continuity at a point and continuity on a set
2.2 - Fundamental theorems on continuity
2.3 - Homeomorphism
2.4 - Limits. Uniqueness limit theorem
2.5 - Sequences. Convergent sequence on a topological space
2.6 - Characterization of the topology by sequences
2.7 - Closure sequence. Sequentially closed set
2.8 - Sequential continuity
3 - Metric Spaces
3.1 - Metric on a set. Definition of metric space
3.2 - Open ball, closed ball and sphere
3.3 - Bounded set. Bounded function
3.4 - Topology associated with the metric
3.5 - Convergence of sequences on a metric space
3.6 - Equivalent metrics
3.7 - Metrizable topological space
3.8 - Characterization of continuity in terms of metrics
3.9 - Uniform continuity
3.10 - Lipschitz function
3.11 - Functional spaces: pointwise convergence and uniform convergence of sequences of functions
3.12 - Topology of uniform convergence
3.13 - Product metrics
3.14 - Inequalities of Young, of Holder to finite sums, of Cauchy-Schwarz to finite sums and of Minkowski to finite sums
4 - Compactness
4.1 - Open covering of a set
4.2 - Property of Heine-Borel-Lebesgue
4.3 - Compact topological space
4.4 - Theorems on compactness: continuous functions on compact spaces
4.5 - Sequentially compact space
4.6 - Locally compact space
4.7 - Alexandroff compactification
4.8 - Compactness on metric spaces
4.9 - Bolzano-Weierstrass theorem
4.10 - Characterization of the compacts sets of IR
5 - Connectedness
5.1 - Disconnected topological space
5.2 - Connected topological space
5.3 - Continuous functions on connected spaces
5.4 - Characterization of the connected sets of IR
5.5 - Path connected space
5.6 - Locally connected space
5.7 - Locally path connected space

# Literature/Sources:

N. Bourbaki , 1990 , Topologie Générale , Masson
M. Hichem Mortad , 2016 , Introductory Topology: Exercises and Solutions , WSPC
C. Gustave , 1966 , Topology , Academic Press
C. Michael , 1990 , Elementary Topology , Dover
J. Kelley , 1975 , General Topology , Springer
L. Loura , 2001 , Curso de Topologia Geral , UMa
J. Munkres , 1975 , Topology: a first course , Prentice-Hall
L. Schwartz , 1971 , Topologie Générale et Analyse Fonctionnelle , Hermann
P. Alexandroff , 1984 , Topologie , Springer

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Oral and written presentation of the syllabus contents of the curricular unit. Discussion and resolution of exercises and application problems in small groups or individually. Realization of two tests (with weight of 50% each) to be solved individually during the normal season. In this way, the student can, during the semester, evaluate their performance and change strategies if necessary. In the resit period of exams, students can perform only one of the tests or, alternatively, the complete exam corresponding to 100% of the final grade. The importance of this examination, in addition to the assessment objectives, focuses on the student's ability to relate different parts of the subject.