Identifying some properties of a set

Question

Let $S \subset R^2$ be defined by

$S$ = {$(m+ \frac{1}{4^{|p|}} , n+ \frac{1}{4^{|q|}}): m,n,p,q \in Z$}

Then,

1. $S$ is discrete in $R^2$.

2. The set of limit points of $S$ is the set {$(m,n) : m,n \in Z$}.

3. $S^c$ is connected but not path connected.

4. $S^c$ is path connected.

This question appeared in CSIR Dec 2015.

I don't have any idea about how to do this question. Please help!

Thanks in advance!

Answer 1

Hint/Solution:

$(1).$ Note that $(m,n)$ is in $S$ and is also limit point of $S$ hence $S$ is not discrete.

$(2).$ Note that elements of the form $(m+ \frac{1}{4^p},n)$ are also limit points of $S$

$(3,4.)$ If A is countable then $\mathbb R^2$-$A$ is path connected