# 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. This page will be updated periodically.

## Short questions

- Is it possible to construct a topology on every set?
- Give an example of open set in $\mathbb{R}$ with usual topology, which is not an open interval.
- Let $X=\{a\}$. Then what are the differences between discrete topology, indiscreet topology and confinite topology on $X$?
- Let $X$ be a non-empty finite set. Then what is the difference between discrete and cofinite toplogy on $X$.
- Let $\tau$ be a cofinite toplogy on $\mathbb{N}$. Then write any three element of $\tau$.
- Let $(\mathbb{Z}, \tau)$ be a cofinite topological spaces.
- Is $\mathbb{N}$ open in $\tau$?
- Is $A=\{\pm 100,\pm 101, \pm 102, \ldots \}$ open in $\tau$?
- Is $E=\{0,\pm 2,\pm 4,\ldots\}$ open in $\tau$?
- Is set of prime open in $\tau$?
- Is $B=\{1,2,3,\ldots,99\}$ closed in $\tau$?
- Is $C=\{10^{10}+n : n \in \mathbb{Z} \}$ open in $\tau$?

- Write the closure of the set $S=\left\{1+\frac{1}{n}: n \in \mathbb{N} \right\}$ in usual topology on $\mathbb{R}$.
- What is the closure of the set $T=\{1,2,3,4,5 \}\cup (6,7) \cup (7,8] $ in usual topology on $\mathbb{R}$?
- What is the closure of the set $U=\{101,102,103,\ldots,200\}$ in a cofinite toplogy constructed on $\mathbb{Q}$?

## Multiple choice questions (MCQs)

- 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) $\tau_1 \backslash \tau_2$
- (D) $\tau_2 \cup \tau_1$

- 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

- 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

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