2.2. Topology of Real Line

Topology of metric spaces is fully developed in Metric Spaces. In this section, we quickly describe the topology of the real line $\mathbb{R}$ as it will be needed for the development of the material related to real functions. We state a number of results without proving. All of these results can be proven in the more general context of metric spaces. Please see Metric Topology for detailed proofs.

2.2.1. Distance

Definition 2.9 (Distance function)

The distance function between two real numbers is defined as

$$d(x,y)=|x-y|\mathrm{\forall }x,y\in \mathbb{R}.$$

Remark 2.5

The distance function satisfies following properties:

1. $d(x,y)\ge 0$.

2. $d(x,y)=0⟺x=y$.

3. $d(x,y)=d(y,x)$.

4. $d(x,y)\le d(x,z)+d(z,x)$ due to Proposition 2.4.

It is a metric.

2.2.2. Neighborhoods or Balls

Definition 2.10 (Neighborhood)

Given a real number $x\in \mathbb{R}$ and $ϵ>0$, the set

$$V_{ϵ}(x)=\{y\in \mathbb{R}||y-x|<ϵ\}$$

is called the $ϵ$-neighborhood of $x$. It is also called an open ball.

In other words, $V_{ϵ}(x)=(x-ϵ,x+ϵ)$. A neighborhood is an open interval.

Definition 2.11 (Closed neighborhood)

Given a real number $x\in \mathbb{R}$ and $ϵ>0$, the set

$$C_{ϵ}(x)=\{y\in \mathbb{R}:|y-x|\le ϵ\}$$

is called the $ϵ$-closed-neighborhood of $x$. It is also called a closed ball.

In other words, $C_{ϵ}(x)=[x-ϵ,x+ϵ]$. A closed neighborhood/ball is a closed interval.

Definition 2.12 (Deleted neighborhood)

Given a real number $x\in \mathbb{R}$ and $ϵ>0$, the set $(x-ϵ,x)\cup (x,x+ϵ)$ is called a deleted-neighborhood of $x$.

The deleted neighborhood doesn’t include $x$.

2.2.3. Open Sets

Definition 2.13 (Open sets)

A subset $A$ of $\mathbb{R}$ is said to be open in $\mathbb{R}$ if for every $x\in A$ there exists an “open ball” entirely within $A$.

In other words, there exists an $r>0$ such that $(x-r,x+r)\subseteq A$.

We claim without proving:

1. Every open ball/neighborhood is an open set.

2. Arbitrary unions of open sets are open sets.

3. Finite intersections of open sets are open sets.

4. Every open interval is an open set.

See Metric Topology for proofs.

There is one interesting result specific to real lines which we prove here.

Theorem 2.1

Let $E\subseteq \mathbb{R}$ be an open set. Then there exist countably many disjoint open intervals $I_j,j=1,2,\dots$ such that

$$E=\bigcup _{j=1}^{\mathrm{\infty }}{I}_j.$$

Proof. We prove this result by establishing an equivalence relation on the set $E$.

1. If $a,b\in E$, we say that $a\sim b$ if the entire open interval $(a,b)$ is contained in $E$; i.e., $(a,b)\subseteq E$.

2. Readers can verify that it is indeed an equivalence relation.

3. This equivalence relation partitions $E$ into a disjoint union of equivalence classes.

4. Let each class be labeled as $I_j$ for $j\in J$ where $J$ is an arbitrary index set. We don’t yet know that $J$ is countable.

5. We can see that $I_j$ must be an interval for every $j\in J$.

   1. Let $a_j,b_j\in I_j$, with $a_j<b_j$.

   2. Then $a_j\sim b_j$.

   3. Hence $(a_j,b_j)\subseteq I_j$.

   4. Since this is true for every $a_j,b_j\in I_j$, hence $I_j$ must be an interval.

6. We next show that $I_j$ must be open for every $j\in J$.

   1. Let $x\in I_j$ be arbitrary.

   2. Since $x\in E$ and $E$ is open, hence there exists $ϵ>0$ such that $(x-ϵ,x+ϵ)\subseteq E$.

   3. Then $a\sim x$ for every $a\in (x-ϵ,x+ϵ)$.

   4. Hence $(x-ϵ,x+ϵ)\subseteq I_j$.

   5. Hence $I_j$ must be open.

7. We next show that $J$ (index set) must be finite or countable.

   1. Since each $I_j$ is an open interval, hence it must contain at least one rational number (Proposition 2.11).

   2. Since there are countably many rational numbers, hence there can be at most countably many $I_j$.

2.2.4. Closed Sets

Definition 2.14 (Closed sets)

A subset $A$ of $\mathbb{R}$ is said to be closed in $\mathbb{R}$ if $\mathbb{R}\setminus A$ is open in $\mathbb{R}$.

We claim without proving:

1. $\varnothing$ and $\mathbb{R}$ are both open and closed subsets of $\mathbb{R}$.

2. Every singleton (or a degenerate interval) is a closed set.

3. A closed neighborhood/ball is a closed set.

4. Every closed interval is a closed set.

5. Half open intervals are neither open nor closed.

6. Arbitrary intersections of closed sets are closed sets.

7. Finite unions of closed sets are closed sets.

8. Any finite set is closed.

9. The set of natural numbers is closed.

10. The set of integers is closed.

2.2.5. Interior

Definition 2.15 (Interior point)

A point $x$ is called an interior point of a set $A\subseteq \mathbb{R}$ if there exists an open interval $(x-r,x+r)$ such that $(x-r,x+r)\subseteq A$.

By definition an interior point of a set belongs to the set too. $x\in (x-r,x+r)\subseteq A$.

Definition 2.16 (Interior)

Let $A\subseteq \mathbb{R}$. The largest open set in $\mathbb{R}$ that is contained in $A$ is called the interior of $A$ and is denoted by $\mathrm{int}A$.

Note that $\mathrm{int}A\subseteq A$.

We claim without proving.

1. Let $O\subseteq A$ be an open set. Then $O\subseteq \mathrm{int}A$.

2. The interior of a set $A$ is the collection of all the interior points of $A$.

3. $A$ is open if and only if $A=\mathrm{int}A$.

2.2.6. Closure

Definition 2.17 (Closure point)

A point $x\in \mathbb{R}$ is called a closure point of a subset $A$ of $\mathbb{R}$ if every open ball/neighborhood at $x$ contains (at least) one point in $A$.

In other words:

$$(x-r,x+r)\cap A\ne \varnothing \mathrm{\forall }r>0.$$

Definition 2.18 (Closure)

Let $A\subseteq \mathbb{R}$. The smallest closed set in $\mathbb{R}$ that contains $A$ is called the closure of $A$ and is denoted by $\mathrm{cl}A$.

We claim without proving:

1. Every point in $A$ is a closure point of $A$.

2. Let $C$ be a closed subset of $\mathbb{R}$ such that $A\subseteq C$. Then $\mathrm{cl}A\subseteq C$.

3. The closure of a set $A$ is the collection of all the closure points of $A$.

4. $A$ is closed if and only if $A=\mathrm{cl}A$.

5. Let $A\subseteq \mathbb{R}$. Then

   $$\mathbb{R}\setminus (\mathrm{int}A)=\mathrm{cl}(\mathbb{R}\setminus A).$$

Example 2.5

Consider $\mathbb{Q}$, the set of rational numbers.

1. Recall that every interval of real numbers contains a rational number (Proposition 2.11).

2. Thus, every neighborhood of a real number contains a rational number.

3. Thus, every real number is a closure point of $\mathbb{Q}$.

4. Thus, $\mathrm{cl}\mathbb{Q}=\mathbb{R}$.

2.2.7. Boundary

Definition 2.19 (Boundary point)

A point $x\in \mathbb{R}$ is called a boundary point of $A$ if every open ball $(x-r,x+r)$ at $x$ contains points from $A$ as well as $\mathbb{R}\setminus A$.

Definition 2.20 (Boundary)

The boundary of a set $A\subseteq \mathbb{R}$, denoted by $\mathrm{bd}A$ is defined as the set of all boundary points of $A$.

We claim without proving:

1. $\mathrm{bd}A=\mathrm{cl}A\setminus \mathrm{int}A$.

2. For intervals $(a,b)$, $(a,b]$, $[a,b)$ and $[a,b]$, the boundary points are $a$ and $b$.

2.2.8. Accumulation

Definition 2.21 (Accumulation point)

A point $x\in \mathbb{R}$ is called an accumulation point of a set $A\subseteq \mathbb{R}$, if every neighborhood of $x$ contains a point in $A$ distinct from $x$.

$$(x-r,x+r)\cap A\setminus \{x\}\ne \varnothing \mathrm{\forall }r>0.$$

In other words, every deleted neighborhood of $x$ contains a point in $A$.

Some authors call accumulation points as limit points. Some authors make a distinction between accumulation points and limit points.

Definition 2.22 (Derived set)

The set of accumulation points of a set $A$ is called its derived set and is denoted by $A^{\prime }$.

Definition 2.23 (Isolated point)

A point $x\in A$ is called isolated if there is a neighborhood of $x$ that doesn’t contain any other point of $A$.

We claim without proving:

1. Every accumulation point is a closure point.

2. A closure point is either an accumulation point or an isolated point.

3. $\mathrm{cl}A=A\cup A^{\prime }$.

4. A set is closed if and only if it contains all its accumulation points.

5. In other words, a set is closed if and only if its complement doesn’t contain any of its accumulation points.

6. A singleton set $\{x\}$ doesn’t have any accumulation points.

7. A set consisting of isolated points doesn’t have any accumulation points.

2.2.9. Exterior

Definition 2.24 (Exterior point)

A point $x$ is called exterior to a set $A\subseteq \mathbb{R}$ if it is interior to $RR\setminus A$.

Definition 2.25 (Exterior)

The set of all exterior points of $A$ is called its exterior.

2.2.10. Open Cover

Definition 2.26 (Open cover)

A collection $\mathcal{O}$ of open sets is called an open cover or open covering of a set $A$ if for every $x\in A$, there exists a set $O\in \mathcal{O}$ such that $x\in O$.

In other words:

$$A\subseteq \bigcup _{O\in \mathcal{O}}O.$$

An open cover is called finite or finite open cover if it consists of finitely many open sets.

A subset of a cover is known as a subcover.

Theorem 2.2 (Heine-Borel theorem)

If $\mathcal{O}$ is an open cover of a closed and bounded subset $A\subseteq \mathbb{R}$, then $A$ has an open cover $\mathcal{P}$ consisting of finitely many open sets belonging to $\mathcal{O}$.

2.2.11. Compact Sets

Definition 2.27 (Compact set)

A set $A\subseteq \mathbb{R}$ is called compact if it is closed and bounded.

We claim without proving:

1. If every open cover of $A$ contains a finite subcover, then $A$ is compact.

Theorem 2.3 (Bolzano-Weierstrass theorem)

Every bounded infinite set of real numbers has at least one limit point.