1. Neighbourhoods of a point in ℝ
Definition
Let $x_{0} \in {\mathbb{R}}$. A set $V \subset {\mathbb{R}}$ is called a neighbourhood of $x_{0}$ if there is $\varepsilon > 0$ such that $\left( x_{0} - \varepsilon,x_{0} + \varepsilon \right) \subset V$. We write $\mathcal{V}\left( x_{0} \right)$ for the set of neighbourhoods of $x_{0}$.
- Equivalently: $V \in \mathcal{V}\left( x_{0} \right) \Leftrightarrow \exists a,b \in {\mathbb{R}}$, $a < x_{0} < b$, such that $(a,b) \subset V$.
Examples. $\left( x_{0} - 1,x_{0} + 1 \right)$, $\left\lbrack x_{0} - 1,x_{0} + 2 \right\rbrack$ and $\mathbb{R}$ are neighbourhoods of $x_{0}$; $\left\lbrack x_{0},x_{0} + 1 \right)$ is not a neighbourhood of $x_{0}$ (it contains no interval centred at $x_{0}$). The set $V = \lbrack 1,3) \cup \left\{ 5 \right\}$ is a neighbourhood of $2$, but not of $1$, $3$ or $5$. The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ are not neighbourhoods of any point.
2. Neighbourhoods of +∞ and −∞
Definitions
- $V \subset {\mathbb{R}}$ is a neighbourhood of $+ \infty$ if there is $a \in {\mathbb{R}}$ such that $(a, + \infty) \subset V$.
- $V \subset {\mathbb{R}}$ is a neighbourhood of $- \infty$ if there is $a \in {\mathbb{R}}$ such that $( - \infty,a) \subset V$.
Examples. $(5, + \infty)$, $\lbrack 0, + \infty)$ and ${\mathbb{R}} \smallsetminus \lbrack 0,1\rbrack$ are neighbourhoods of $+ \infty$; $\mathbb{N}$ is not a neighbourhood of $+ \infty$.
3. Properties of neighbourhoods
Let $x_{0} \in \overline{\mathbb{R}}$.
(1) If $x_{0} \in {\mathbb{R}}$ and $V \in \mathcal{V}\left( x_{0} \right)$, then $x_{0} \in V$.
(2) If $V \in \mathcal{V}\left( x_{0} \right)$ and $V \subset W$, then $W \in \mathcal{V}\left( x_{0} \right)$.
(3) If $V_{1},V_{2} \in \mathcal{V}\left( x_{0} \right)$, then $V_{1} \cap V_{2} \in \mathcal{V}\left( x_{0} \right)$. Proof ($x_{0} \in {\mathbb{R}}$): if $\left( x_{0} - \varepsilon_{1},x_{0} + \varepsilon_{1} \right) \subset V_{1}$ and $\left( x_{0} - \varepsilon_{2},x_{0} + \varepsilon_{2} \right) \subset V_{2}$, then for $\varepsilon = \min\left\{ \varepsilon_{1},\varepsilon_{2} \right\}$ we have $\left( x_{0} - \varepsilon,x_{0} + \varepsilon \right) \subset V_{1} \cap V_{2}$.
(4) Separation property: if $x_{0},y_{0} \in \overline{\mathbb{R}}$, $x_{0} \neq y_{0}$, there are $U \in \mathcal{V}\left( x_{0} \right)$ and $W \in \mathcal{V}\left( y_{0} \right)$ with $U \cap W = \varnothing$. (For $x_{0},y_{0} \in {\mathbb{R}}$ take $\varepsilon = \frac{\left| x_{0} - y_{0} \right|}{2}$.)
4. Accumulation points. Isolated points
Let $A \subset {\mathbb{R}}$ and $x_{0} \in \overline{\mathbb{R}}$.
Definitions
- $x_{0}$ is an accumulation point of the set $A$ if every neighbourhood of $x_{0}$ contains at least one point of $A$ different from $x_{0}$:
$$V \cap \left( A \smallsetminus \left\{ x_{0} \right\} \right) \neq \varnothing,\mkern9mu\forall V \in \mathcal{V}\left( x_{0} \right).$$
For $x_{0} \in {\mathbb{R}}$: $x_{0} \in A' \Leftrightarrow \forall\varepsilon > 0,\mkern9mu\left( x_{0} - \varepsilon,x_{0} + \varepsilon \right) \cap \left( A \smallsetminus \left\{ x_{0} \right\} \right) \neq \varnothing$.
- $A'$ – the set of accumulation points of $A$ (the derived set of $A$).
- A point $a \in A$ that is not an accumulation point of $A$ is called an isolated point of $A$: there is $V \in \mathcal{V}(a)$ with $V \cap A = \left\{ a \right\}$.
Examples.
- $A = (0,1\rbrack$: $A' = \lbrack 0,1\rbrack$; $0 \in A'$, although $0 \notin A$; $A$ has no isolated points.
- $A = {\mathbb{N}}$: $A' = \left\{ + \infty \right\}$; all points of $\mathbb{N}$ are isolated.
- $A = \left\{ \frac{1}{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}}^{*} \right\}$: $A' = \left\{ 0 \right\}$, because for every $\varepsilon > 0$ there is $n > \frac{1}{\varepsilon}$, so $\frac{1}{n} \in ( - \varepsilon,\varepsilon)$; all points of $A$ are isolated.
- $A = (0,1) \cup \left\{ 2 \right\}$: $A' = \lbrack 0,1\rbrack$, and $2$ is an isolated point.
Remarks: a finite set has no accumulation points; $+ \infty \in A' \Leftrightarrow A$ is not bounded above; $- \infty \in A' \Leftrightarrow A$ is not bounded below. The limit of a function $f:D \rightarrow {\mathbb{R}}$ will be studied at the accumulation points of $D$.