Final answer:
A complete metric space is a mathematical structure where every Cauchy sequence has a limit within the same space, meaning it has no "holes" and supports robust analysis within the context of topology and calculus.
Step-by-step explanation:
A complete metric space is a crucial concept in mathematics, particularly in the field of topology and analysis. To define it, let's recall what a metric space is: A metric space is a set equipped with a metric, which is a function that defines a distance between any two elements of the set. To say that a metric space is complete means that every Cauchy sequence of points in the space has a limit that is also within that space. A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. In more intuitive terms, a complete metric space is one where there are no "holes" or "missing points" -- all points that feel like they should be in the space, based on the distances defined by the metric, actually are in the space.
When considering applications, the concept of completeness can have profound implications. For example, the real numbers ℝ with the usual metric (the absolute value of the difference) form a complete metric space. This is fundamental for the rigorous development of calculus and real analysis.