Metric spaces

Course information

Learning outcomes

After passing the course the student knows

- methods of rigorous reasoning in mathematical analysis

- basic topology of inner product spaces, normed spaces, and metric spaces
- the notions of limit and continuity
- the definitions and fundamental properties of compactness, completeness, and connectedness in metric spaces.

Content

Real numbers, metric, norm, inner product, open and closed sets, continuous mappings, sequences and limits, compactness, completeness, connectedness.

Teacher

Pekka Alestalo, Sari Rogovin

Study material

W. Rudin: Principles of Mathematical Analysis, 3rd Edition, McGraw-Hill 1976

N.L. Carothers: Real Analysis, Cambridge University Press 2000

J. Väisälä: Topologia I, Limes ry 1999 (in Finnish/Swedish)

Workload

24+12 (4+2).

Prerequisites

Recommended prerequisites: MS-A01XX

Evaluation

Lectures, exercises and course exam OR exam only

Teaching time and location

The teaching time and the location of the course can be found at Sisu information system

1. The link leads to Completion methods tab of the course page
2. From the top of the page (Version drop-down menu), check that academic year 2025-2026 is selected  

3. From the blue bar, click Lecture and open the Groups and teaching times tab to see the teaching time and the location of the course. Exam times can be seen by clicking Exam.
    

   Exam instructions at the Open University

The teacher has four (4) weeks to grade the exam submissions starting from the exam date.

Further information

Substitutes the course Mat-1.2990 Foundations of Modern Analysis, and the course MS-C1540 Euclidean spaces.

Online learning environment for the course  

MyCourses online learning environment is a tool for both students and teachers for communication and managing everyday course work. You need an Aalto User ID to log in to the course's MyCourses workspace and participate in teaching.

Digital workspace of the course:
MyCourses

More details on completing the course will be available at MyCourses closer to the start of the course.

Registration and course fee

Registration for Open University courses is done via the link below: 
Course registration

Registration for this course starts on 8.12.2025 at 9.00  
Registration for this course ends on 12.1.2026 at 23.59

It is not possible to register before the registration period has started or after it has ended.

The number of participants is limited to 5.

This course is organized by the Aalto University School of Science.

The course fee is paid upon registration for each course at a time and it is binding. Familiarize yourself with registration and payment rules as well as other guidelines for students:
Registration, payments and rules

Activating Aalto ID and Multifactor Authentication

You can activate your Aalto User ID the day after the registration and you should activate it no later than the day before the start of the course.
Instructions for activating your Aalto user ID

Once you have activated your Aalto user ID, also enable multifactor authentication. Some of Aalto University's services require multifactor authentication.

Instructions for using Multifactor Authentication