Topology MCQs Math
\[
\textbf{MCQs on Topology with Answers}
\]
\[
\textbf{Q1: Which of the following is a topological space?}
\]
\[
\text{(A) } \mathbb{R}^2 \quad
\text{(B) } \mathbb{N} \quad
\text{(C) } \mathbb{Z} \quad
\text{(D) } \mathbb{Q}
\]
\[
\textbf{Answer: (A) } \mathbb{R}^2
\]
\[
\textbf{Q2: A subset } A \text{ of a topological space } X \text{ is open if:}
\]
\[
\text{(A) } A \text{ is the union of open sets} \quad
\text{(B) } A \text{ is the intersection of closed sets} \quad
\text{(C) } A \text{ is the union of closed sets} \quad
\text{(D) } A \text{ contains at least one point}
\]
\[
\textbf{Answer: (A) } A \text{ is the union of open sets}
\]
\[
\textbf{Q5: The space } \mathbb{R} \text{ with the standard topology is:}
\]
\[
\text{(A) } Connected \quad
\text{(B) } Disconnected \quad
\text{(C) } Neither connected nor disconnected \quad
\text{(D) } Isolated
\]
\[
\textbf{Answer: (A) } Connected
\]
\[
\textbf{Q6: Which of the following is a basis for the topology on } \mathbb{R} \text{?}
\]
\[
\text{(A) } \{(a, b) \ | \ a, b \in \mathbb{R} \text{ and } a < b \} \quad
\text{(B) } \{(a, b) \ | \ a, b \in \mathbb{R} \text{ and } a > b \} \quad
\text{(C) } \{[a, b] \ | \ a, b \in \mathbb{R} \text{ and } a < b \} \quad
\text{(D) } \{(a, b) \ | \ a, b \in \mathbb{R} \text{ and } a = b \}
\]
\[
\textbf{Answer: (A) } \{(a, b) \ | \ a, b \in \mathbb{R} \text{ and } a < b \}
\]
\[
\textbf{Q7: If a function is continuous on a compact set, then it is:}
\]
\[
\text{(A) } Always differentiable \quad
\text{(B) } Always bounded \quad
\text{(C) } Always injective \quad
\text{(D) } Always surjective
\]
\[
\textbf{Answer: (B) } Always bounded
\]
\[
\textbf{Q9: A topological space is said to be Hausdorff if:}
\]
\[
\text{(A) } Every pair of distinct points has disjoint open neighborhoods \quad
\text{(B) } Every set is open \quad
\text{(C) } Every set is closed \quad
\text{(D) } Every set has a unique boundary
\]
\[
\textbf{Answer: (A) } Every pair of distinct points has disjoint open neighborhoods
\]
\[
\textbf{Q10: The fundamental group of a circle is:}
\]
\[
\text{(A) } \mathbb{Z} \quad
\text{(B) } \mathbb{R} \quad
\text{(C) } \mathbb{Z}_2 \quad
\text{(D) } The trivial group
\]
\[
\textbf{Answer: (A) } \mathbb{Z}
\]