Question

Which of the following statement is true:

• A. Continuous image of a connected set is connected
• B. The union of two connected sets, having non-empty intersection, may not be a connected set
• C. The real line \(\mathbb{R}\) is not connected
• D. A non-empty subset X of \(\mathbb{R}\) is not connected if X is an interval or a singleton set

Hint:

For topology questions, remembering which properties are preserved under continuous mappings is crucial. The most important ones are compactness and connectedness. A continuous function maps a compact set to a compact set and a connected set to a connected set.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests fundamental theorems and definitions related to connected sets in topology. A connected set is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets.

Step 2: Detailed Explanation:
Let's analyze each statement:

(A) Continuous image of a connected set is connected.
This is a fundamental theorem in topology. If \(f: X \to Y\) is a continuous function and the domain \(X\) is a connected set, then its image \(f(X)\) is also a connected set in \(Y\). This statement is true.

(B) The union of two connected sets, having non-empty intersection, may not be a connected set.
This statement is false. A standard theorem states that if \(A\) and \(B\) are connected sets and their intersection \(A \cap B\) is non-empty, then their union \(A \cup B\) is also connected. The non-empty intersection ensures there is no "gap" between the two sets.

(C) The real line \(\mathbb{R}\) is not connected.
This statement is false. The real line \(\mathbb{R}\) with its standard topology is the quintessential 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. In fact, the converse is true: a non-empty subset of \(\mathbb{R}\) is connected if and only if it is an interval. A singleton set \(\{a\}\) is considered a trivial interval \([a, a]\) and is connected.

Step 3: Final Answer:
Based on the analysis of the fundamental properties of connected sets, the only true statement is (A).