Step 1: Core Concept:
This question assesses understanding of connected sets in topology. A connected set is a topological space that cannot be expressed as the union of two or more separate, non-empty open subsets.
Step 2: Statement Analysis:
(A) A continuous image of a connected set is connected.
This is a key theorem. If \(f: X \to Y\) is continuous, and \(X\) is connected, then its image \(f(X)\) is also connected. This statement is true.
(B) The union of two connected sets, with a non-empty intersection, may not be connected.
This statement is false. The union of two connected sets \(A\) and \(B\), where their intersection \(A \cap B\) is non-empty, is also connected. The overlap prevents a "separation".
(C) The real line \(\mathbb{R}\) is not connected.
This statement is false. The real line \(\mathbb{R}\) with its standard topology is a fundamental example of a connected set.
(D) A non-empty subset X of \(\mathbb{R}\) is not connected if X is an interval or a singleton set.
This statement is false. A non-empty subset of \(\mathbb{R}\) is connected if and only if it is an interval. A singleton set \(\{a\}\) is a trivial interval \([a, a]\) and is connected.
Step 3: Conclusion:
Based on the analysis, only statement (A) is true.