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Let $S\subset \mathbb{R^2}$ be defined by $S=\{(m+\frac{1}{4^{|p|}}, n+\frac{1}{4^{|q|}}):m,n,p,q \in \mathbb{Z}\}$. . Then,

  1. S is discrete in $\mathbb R^2$ .
  2. The set of limit points of S is the set {(m,n);m,n $\in \mathbb Z$}.

i don't understand this

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  • $\begingroup$ If A is the set of limit points of S, then S⊆A ?? in this link... $\endgroup$ Aug 3, 2017 at 16:41

2 Answers 2

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2)as p and q tend to $\infty$ & fix p and let q tend to $\infty$ & fix q and let p tend to $\infty$ so it doesn't contain all limit points so b option incorrect.

1)there isn't given any topology (so i'm not sure what i'm thinking is correct or not,since i'm new in this)

since S contain some limit points so some points aren't isolated that's why it's not discrete.

is my approch correct...?

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Discrete set: Discrete set is a set of points of a topological space such that each point in the set is an isolated point, i.e. a point that has a neighborhood that contains no other points of the set.

Clearly, $(5/4,1)$ is a limit point of $S$ and $(5/4,1)\not\in \{(m,n);m,n ∈\mathbb Z\}$.
Also $(1,1)\in S$ and $(1,1)$ is limit point as $(1,1)=(0+1/4^0, 0+1/4^0)$.
Thus $S$ is not discrete set.
Hence both the two options are incorrect.

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