====== Topology: Short Questions and MCQs ======

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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]]. This page will be updated periodically. 
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===== 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 $A=\{\pm 100,\pm 101, \pm 102, ... \}$ 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 $\tau$?
  - 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 $U=\{101,102,103,...,200\}$ in a cofinite toplogy constructed on $\mathbb{Q}$?

===== Multiple choice questions (MCQs) =====
<panel>
  - If $\tau_1$ and $\tau_2$ are two typologies on non-empty set $X$, then ................... is topological space.\\
    - $\tau_1\cup \tau_2$
    - $\tau_1\cap \tau_2$
    - $\tau_1 \backslash \tau_2$
    - $\tau_2 \cup \tau_1$
</panel>
<panel>
  - If $\tau$ is typology on non-empty set $X$, then arbitrary ................... of member of $\tau$ belong to $\tau$.
    - union
    - intersection
    - product
    - compliment
</panel>
  - If $\tau$ is typology on non-empty set $X$, then arbitrary ................... of member of $\tau$ belong to $\tau$.
    - union
    - intersection
    - product
    - compliment