Let's analyze the statement given in the image **step by step**: --- ### Step 1: Understanding the Statement **Lemma 02.1:** Let \( S \) be a subset of a metric space \( (X, d) \). Then the following conditions are equivalent. *(Only condition (c) is visible):* (c) For every \( \varepsilon > 0 \), there is a finite cover of \( S \) by subsets of \( X \) each with diameter less than \( \varepsilon \). **Explanation:** This condition is describing a property of the set \( S \): for any positive number \( \varepsilon \), you can cover \( S \) using only finitely many subsets of \( X \), and each of these subsets has a "diameter" (the greatest distance between any two points in the subset) less than \( \varepsilon \). --- ### Step 2: Identifying the Concept **Key Concept:** This property is the definition of **total boundedness** in a metric space. **Explanation:** A subset \( S \) of a metric space \( (X, d) \) is said to be *totally bounded* if for every \( \varepsilon > 0 \), \( S \) can be covered by finitely many subsets of diameter less than \( \varepsilon \) (often, these subsets are taken to be open balls, but the definition here uses general subsets). --- ## **Final Answer:** **Condition (c) is the definition of a set being *totally bounded* in a metric space.**