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. The author of this page is Dr. Atiq ur Rehman, PhD. 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 $\mathbb{R}$ with usual topology, which is not an open interval.
  3. Let $X=\{a\}$. 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 $\mathbb{N}$. Then write any three element of $\tau$.
  6. Let $(\mathbb{Z}, \tau)$ be a cofinite topological spaces.
    • Is $\mathbb{N}$ open in $\tau$?
    • Is $A=\{\pm 100,\pm 101, \pm 102, ... \}$ open in $\tau$?
    • Is $E=\{0,\pm 2,\pm 4,...\}$ open in $\tau$?
    • Is set of prime open in $\tau$?
    • Is $B=\{1,2,3,...,99\}$ closed in $\tau$?
    • Is $C=\{10^{10}+n : n \in \mathbb{Z} \}$ open in $\tau$?
  7. Write the closure of the set $S=\left\{1+\frac{1}{n}: n \in \mathbb{N} \right\}$ in usual topology on $\mathbb{R}$.
  8. What is the closure of the set $T=\{1,2,3,4,5 \}\cup (6,7) \cup (7,8] $ in usual topology on $\mathbb{R}$?
  9. What is the closure of the set $U=\{101,102,103,...,200\}$ in a cofinite toplogy constructed on $\mathbb{Q}$?

Multiple choice questions (MCQs)

  1. If $\tau_1$ and $\tau_2$ are two typologies on non-empty set $X$, then ………………. is topological space.
    1. $\tau_1\cup \tau_2$
    2. $\tau_1\cap \tau_2$
    3. $\tau_1 \backslash \tau_2$
    4. $\tau_2 \cup \tau_1$
  1. If $\tau$ is typology on non-empty set $X$, then arbitrary ………………. of member of $\tau$ belong to $\tau$.
    1. union
    2. intersection
    3. product
    4. compliment
  1. If $\tau$ is typology on non-empty set $X$, then arbitrary ………………. of member of $\tau$ belong to $\tau$.
    1. union
    2. intersection
    3. product
    4. compliment