Topology: Short Questions and MCQs

We are going to add short questions and MCQs for Topology. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. This page will be updated periodically.

Short questions

1. Is it possible to construct a topology on every set?
2. Give an example of open set in $R$ with usual topology, which is not an open interval.
3. Let $X=\left\{a\right\}$. Then what are the differences between discrete topology, indiscreet topology and confinite topology on $X$?
4. Let $X$ be a non-empty finite set. Then what is the difference between discrete and cofinite toplogy on $X$.
5. Let $\tau$ be a cofinite toplogy on $N$. Then write any three element of $\tau$.
6. Let $(Z,\tau )$ be a cofinite topological spaces.
   • Is $N$ open in $\tau$?
   • Is $A=\{\pm 100,\pm 101,\pm 102,\dots \}$ open in $\tau$?
   • Is $E=\{0,\pm 2,\pm 4,\dots \}$ open in $\tau$?
   • Is set of prime open in $\tau$?
   • Is $B=\{1,2,3,\dots ,99\}$ closed in $\tau$?
   • Is $C=\{{10}^{10}+n:n\in Z\}$ open in $\tau$?
7. Write the closure of the set $S=\left\{1+\frac{1}{n}:n\in N\right\}$ in usual topology on $R$.
8. What is the closure of the set $T=\{1,2,3,4,5\}\cup (6,7)\cup (7,8]$ in usual topology on $R$?
9. What is the closure of the set $U=\{101,102,103,\dots ,200\}$ in a cofinite toplogy constructed on $Q$?

Multiple choice questions (MCQs)

1. If ${\tau }_1$ and ${\tau }_2$ are two typologies on non-empty set $X$, then ………………. is topological space.
   • (A) ${\tau }_1\cup {\tau }_2$
   • (B) ${\tau }_1\cap {\tau }_2$
   • (C) ${\tau }_1\setminus {\tau }_2$
   • (D) ${\tau }_2\cup {\tau }_1$
2. If $\tau$ is typology on non-empty set $X$, then arbitrary ………………. of member of $\tau$ belong to $\tau$.
   • (A) union
   • (B) intersection
   • (C) product
   • (D) compliment
3. If $\tau$ is typology on non-empty set $X$, then arbitrary ………………. of member of $\tau$ belong to $\tau$.
   • (A) union
   • (B) intersection
   • (C) product
   • (D) compliment