Conclusion
Following completion of this free OpenLearn course, Metric spaces and continuity, as well as being able to understand the terms and definitions, and use the results introduced, you should also find that your skills and confidence in reading, understanding and writing mathematical arguments are improving.
You should now be able to:
- understand the Euclidean distance function on Rn and appreciate its properties, and state and use the Triangle and Reverse Triangle Inequalities for the Euclidean distance function on Rn
- explain the definition of continuity for functions from Rn to Rmand determine whether a given function from Rn to Rm is continuous
- explain the geometric meaning of each of the metric space properties (M1) – (M3) and be able to verify whether a given distance function is a metric
- distinguish between open and closed balls in a metric space and be able to determine them for given metric spaces
- define convergence for sequences in a metric space and determine whether a given sequence in a metric space converges
- state the definition of continuity of a function between two metric spaces.
This free OpenLearn course is an extract from the Open University course M303 Further pure mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] .
If you feel you are ready to move on in your study of pure mathematics but don’t have time to study a full Open University course at this time, you might like to study one of the free OpenLearn courses Number theory, Group theory and Rings and polynomials. They are further adapted extracts from the Open University course M303 Further pure mathematics.
If you are interested in applied mathematics, you might like to study the free OpenLearn courses Introduction to the calculus of variations, Linear programming – the basic ideas and Kinematics of fluids.
If you are interested in statistics, you might like to study the free OpenLearn courses Modelling events in time, Univariate continuous distribution theory and Point estimation.