Remember
- $V$ is a neighbourhood of $x_{0} \in {\mathbb{R}}$ if $\exists\varepsilon > 0$ such that $\left( x_{0} - \varepsilon,x_{0} + \varepsilon \right) \subset V$.
- $V$ is a neighbourhood of $+ \infty$ if $\exists a \in {\mathbb{R}}$: $(a, + \infty) \subset V$; of $- \infty$ if $\exists a \in {\mathbb{R}}$: $( - \infty,a) \subset V$.
- $x_{0} \in \overline{\mathbb{R}}$ is an accumulation point of $A$ if every neighbourhood of $x_{0}$ contains a point of $A$ different from $x_{0}$. $A'$ = the set of accumulation points.
- $a \in A$ is an isolated point of $A$ if $a \notin A'$.
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Decide which of the following sets are neighbourhoods of the point $x_{0} = 2$; justify.
a) $V_{1} = (1,3)$;
b) $V_{2} = \lbrack 2,5\rbrack$;
c) $V_{3} = (0,4\rbrack \smallsetminus \left\{ 3 \right\}$;
d) $V_{4} = {\mathbb{Z}}$;
e) $V_{5} = \left( - \infty,2 + \frac{1}{100} \right)$;
f) $V_{6} = {\mathbb{R}} \smallsetminus \left\{ 2 \right\}$.
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Which of the following sets are neighbourhoods of $+ \infty$?
a) $(5, + \infty)$;
b) $\mathbb{N}$;
c) $\lbrack - 3, + \infty) \smallsetminus \left\{ 10 \right\}$;
d) ${\mathbb{R}} \smallsetminus \lbrack 0,1\rbrack$;
e) $( - \infty,100)$.
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Find the set of accumulation points $A'$ and the isolated points of the set $A$:
a) $A = \lbrack - 1,3)$;
b) $A = (0,1) \cup \left\{ 2,3 \right\}$;
c) $A = \left\{ \frac{1}{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}}^{*} \right\}$;
d) $A = {\mathbb{Z}}$;
e) $A = \left\{ 1,2,3 \right\}$.
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Find $a \in {\mathbb{R}}$ for which the set $V = (a - 2,2a + 1)$ is a neighbourhood of $3$.
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(optional) Let $A = \left\{ \frac{( - 1)^{n}}{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}}^{*} \right\}$. Show, using the definition, that $0$ is an accumulation point of $A$ and find $A'$.