Answer to Question #175360 in Real Analysis for Anand

Question

Give an example for each of the following.

i) A set in R with a unique limit point.

ii) A set in R whose all points except the one are its limit points.

iii) A set having no limit point.

iv) A set S with S^°= S̅

v) A bijection from N_odd to Z

Answer 1

i) $\{\frac{1}{n}:n\in \mathbb{N}\}\cup\{0\}$ has 0 as only limit point.

$ii)[0,1]\cup \{2\}$ has all points except 2 as its limit point.

iii) $\{1,2\}$ has no limit point.

iv) S=$(-1,1)$ the open interval as a subset of $(-1,1)\cup (5,\infty)$ in subspace topology.

v)$f:\mathbb{N}_{odd}\longrightarrow \mathbb{Z}$ is given by

$8n+1\mapsto 2n$ ,

$8n+3\mapsto -2n$ ,

$8n+5\mapsto 2n+1,$

$8n+7\mapsto -2n+1.$ This clearly a bijection.