Metric spaces and continuity

The key idea is to use three particular properties of the Euclidean distance as the basis for defining what is meant by a general distance function, a metric. Section 1 introduces the idea of a metric space and shows how this concept allows us to generalise the notion of continuity. Section 2 develops the idea of sequences and convergence in metric spaces. Section 3 builds on the ideas from the first two sections to formulate a definition of continuity for functions between metric spaces.

Course learning outcomes

After studying this course, you should 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 Rm and 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.