Let's break down the concepts involved in this question about metric spaces and sequences.

Understanding Metric Spaces and Sequences

A metric space is a set where a distance (or metric) is defined between any two points. We denote a metric space as \( (E, d) \), where \( E \) is the set and \( d \) is the metric function.

Inside a metric space, we can talk about sequences of points.

• Cauchy Sequence: A sequence \( (x_n) \) in a metric space \( (E, d) \) is called a Cauchy sequence if for every positive number \( \epsilon > 0 \), there exists a positive integer \( N \) such that for all integers \( m, n > N \), the distance between \( x_m \) and \( x_n \) is less than \( \epsilon \). In simple terms, the terms of a Cauchy sequence get arbitrarily close to each other as the sequence progresses. Mathematically, this is written as \( d(x_m, x_n) < \epsilon \) for all \( m, n > N \).
• Convergent Sequence: A sequence \( (x_n) \) in a metric space \( (E, d) \) is called a convergent sequence if there exists a point \( x \in E \) such that for every positive number \( \epsilon > 0 \), there exists a positive integer \( N \) such that for all integers \( n > N \), the distance between \( x_n \) and \( x \) is less than \( \epsilon \). In simple terms, the terms of a convergent sequence get arbitrarily close to a specific point \( x \in E \) as the sequence progresses. This point \( x \) is called the limit of the sequence. Mathematically, this is written as \( d(x_n, x) < \epsilon \) for all \( n > N \).

An important property is that every convergent sequence is a Cauchy sequence. However, the converse is not always true in any arbitrary metric space.

Defining a Complete Metric Space

The question describes a metric space \( (E, d) \) where every Cauchy sequence in \( (E, d) \) is convergent. This specific property is the definition of a complete metric space.

So, if in a metric space \( (E, d) \), it is guaranteed that every Cauchy sequence converges to a point within the space \( E \), then the space is called a complete metric space.

Analyzing the Options

Let's look at the given options:

• incomplete: An incomplete metric space is one that is not complete. This means there exists at least one Cauchy sequence in the space that does not converge to a point within that space. This contradicts the condition given in the question.
• complete: As discussed above, the definition of a complete metric space is exactly the property stated in the question: every Cauchy sequence converges within the space.
• semi-complete: The term "semi-complete" is not a standard term used to describe this specific property of metric spaces in typical real analysis or topology contexts.
• none of these: Since "complete" accurately describes the metric space based on the given condition, this option is incorrect.

Therefore, a metric space \( (E, d) \) in which every Cauchy sequence converges is by definition a complete metric space.

This property of being a complete metric space is crucial in many areas of mathematics, especially analysis, as it ensures that limits of Cauchy sequences exist within the space, which is important for defining concepts like continuity and integration.