Topological Properties of Complex Numbers

Q: Q: 2. What do you mean by neighbourhood?

Ans: A delta or \delta-neighbourhood of a point z_0 is the set of all points z such that |z - z_0| < \delta. where \delta is any given positive number. For nbd. of a point z_0 we use the notation N_\delta(z_0) thus N_\delta(z_0) = \{z \in \mathbb{C} : |z - z_0| < \delta \} If z_0 is deleted from the neighbourhood of the point z_0 then it is called deleted \delta-nbd which is denoted as N'_\delta(z_0). Thus N'_\delta(z_0) = \{z \in \mathbb{C} : 0 < |z - z_0| < \delta \} Obviously \delta-nbd of z_0 is the whole region lying inside a circle in Argand diagram with centre at z_0 and radius \delta i.e. an open disk in complex plane with centre at z_0 and radius \delta.

Q: Q: 3. Define the Limit Point

Ans: A point z_0 is called a limit point, cluster point, or point of accumulation of a point set S if every deleted nbd of z_0 contains at least one point of S. i.e. if N'_\delta(z_0) \cap S \neq \phi. Since \delta can be any positive number it follows that if z_0 is a limit point of S, then every nbd of z_0 contains infinitely many points of S. Note that z_0 may or may not belong to the set S. By answering the above questions,you can know about the primary terms of Topological Properties .

Sanjay Kumawat October 25, 2026 Complex Analysis(English) No Comments

Contents hide

1 1.Topological Properties of Complex Numbers

1.1 2.Topological Properties

1.2 3.Frequently Asked Questions Related to Topological Properties of Complex Numbers

1.2.1 Q:1.What is a point set?

1.2.2 Q: 2. What do you mean by neighbourhood?

1.2.3 Q: 3. Define the Limit Point

1.2.4 Also Read This Article:- Stereographic Projection and Inverse Points of Complex Numbers

1.2.5 4.छात्र-छात्राओं से आज का प्रश्न (Today’s Question to Students):

1.Topological Properties of Complex Numbers

Topological Properties of Complex Numbers

Learn about Topological Properties of Complex Numbers with clear steps,definitions,theorems for mathematics students.

Also Read This Article:- Topological property

2.Topological Properties

Topological Properties of Complex Numbers

Topological properties of Complex Numbers
Learn how Topological properties of complex numbers with clear steps, Definitions, Theorems for mathematics students.
Topological properties
(1.) Interior, Exterior and Boundary points:
Limits points are of two types, the interior point and boundary point. A point $z_0$ is called an interior point of a set S if we can find as neighbourhood of all of whose points belong to S i.e. which is contained in S. Thus $z_0$ is an interior point $\iff \exists r > 0$ s.t. $N'_{\delta}(z_0) \subset S$ .
If every $\delta$ -nbd of $z_0$ contains points belonging to S and also points not belonging to S, then $z_0$ is called the boundary point of S i.e., $z_0$ is a boundary point of S.
$\iff \exists r > 0, N_{\delta}(z_0) \not\subset S \text{ and } N_{\delta}(z_0) \cap S \neq \phi$
Again $z_0$ is called an exterior point of a set S if it is neither an interior point nor a boundary point of S.
For example all the points lying inside a circle $|z| = r$ are interior points of $|z| < r$ while the points lying on circumference of the circle are boundary points of the set $|z| < r$ . Also zero is a boundary point of the point set $0 < |z| < r$
(2.) Derived and closure of a set
The collection of all limit points of the set S is called a derived set of S and is denoted by the symbol $S'$ . Again the union of S and $S'$ is called the closure of the set S which is denoted by $\bar{S}$ .
(3.) Closed set
If all limit points (interior and boundary points) of a set S belong to S, then S is called a closed set, i.e., if $S' \subset S$ or $\bar{S} = S$ .
Obviously $S'$ is the smallest closed set containing S.
For example ${z : |z| \le 1}$ is a closed set because all the limit points of this set belong to the set.
(4.) Open set: An open set is a set which consists only of interior points i.e. for all $z_0 \in S$ $\exists$ at least one nbd $N_{\delta}(z_0)$ s.t. $N_{\delta}(z_0) \subset S$ .
For example $S = \{z : |z| < 1 \}$ is an open set.
In fact every open disk is an open set.
Also $\mathbb{C}$ and $\phi$ are open sets.
(5.) Bounded set: A set S is called bounded if we can find a positive real number $k$ s.t. $\dots$
(6.) Connected set:
Definition: An open set S is $\dots$
said to be connected if any two points of the set can be joined by a curve consisting of straight line segments (i.e. a polygon curve) all points of which are in S. Thus S is a connected set if it is not the union of two disjoint open (or closed) sets, i.e. there do not exist two open (or closed) sets A and $B$ so that $A \cap B = \phi$ and $A \cup B = S$ . If a set S is not connected then it is called a disconnected set.
(7.) Line segment of an interval: If $z_1$ and $z_2$ are any two points in the complex plane, then the set $\{z : z = z_1 + t(z_2 - z_1), , 0 \leq t \leq 1 \}$ represents a line segment joining points $z_1$ and $z_2$ in complex plane which is denoted as $[z_1, z_2]$ . By the definition of connectedness it is clear that every line segment is a connected subset of C.
(8.) Polygon: If $z_1, z_2, \dots, z_n$ are points in a complex plane, then the set $P = [z_1, z_2] \cup [z_2, z_3] \cup \dots \cup [z_{n-1}, z_n]$ is called a polygon in the complex plane whose end points are $z_1$ and $z_n$ . Line segments $[z_{i-1}, z_i], , i = 1, \dots, n$ are called edges or sides of the polygon. This polygon is denoted by the symbol $P = [z_1, z_2, \dots, z_n]$ . If $z_1 = z_n$ , then the polygon is said to be a closed polygon. Indeed every polygon is a connected subset of C.

Topological Properties of Complex Numbers

Theorem: A nonvoid open subset S of C is connected if for any two points $a, b$ in S, there exists a polygon.
from a polygon lying entirely inside S.
Proof: Suppose that S satisfies this condition and let us assume that S is not connected. Then there exists two non-empty open sets A and $B$ such that
$S = A \cup B, , A \cap B = \phi$
and A and $B$ are both open and closed.
Let $a \in A$ and $b \in B$ ; by hypothesis there is a polygon $P$ from $a$ to $b$ such that $P \subset S$ . Now a moment thought will show that one of the segments making up $P$ will have one point in A and another in $B$ . So we can assume that $P = [a, b]$ . Define
$X = {s \in [0, 1] : sb + (1 - s)a \in A}$
$Y = {t \in [0, 1] : tb + (1 - t)a \in B}$
Then $X$ and $Y$ are non-empty open subsets of $[0, 1]$ such that $X \cap Y = \phi$ and $X \cup Y = [0, 1]$ . This means that $[0, 1]$ is not connected, which is a contradiction. Thus, S must be connected.
Conversely, assume that S is a connected set. For a fixed $a \in S$ , let A be the set of all those points of S which can be joined to $a$ by a polygon lying entirely in S i.e.
$A = \{b \in S : \text{there is a polygon } P \subset S \text{ from } a \text{ to } b \}$
and let $B = S \sim A$ . Then $B$ is the set of those points of S which cannot be joined to $a$ by a polygon lying entirely in S. Clearly $A \cap B = \phi$ . The set A is non-empty since it contains the point $a$ . We now prove that A is open set in S.
Let $b \in A$ and let $P = {a, z_1, \dots, z_n, b}$ be a polygon from $a$ to $b$ with $P \subset S$ . Since S is open, therefore for $b \in A$ there exists an $\epsilon > 0$ such that $N_\epsilon(b) \subset S$ . But if $z \in N_\epsilon(b)$ then $[b, z] \subset N_\epsilon(b) \subset S$ . Hence the polygon $Q = P \cup [b, z]$ lies completely in S and joins $a$ to $z$ . This shows that $N_\epsilon(b) \subset A$ and so A is open. Similarly we can prove that $B$ is open subset in S.
(9.) Open domain or Region
Definition: A non-empty open connected subset in a complex plane is called an domain or an open region. The term domain is also common but we shall avoid its usage as it has other usages. However, we shall invariably employ the symbol $G$ for a region in $\mathbb{C}$ . The closure of an open region or domain is called a closed region.
(10.) Sequence
Definition: A sequence of complex numbers is a rule of which assigns to each natural number $n$ , a complex number $z$ . Thus $f: \mathbb{N} \to \mathbb{C}$ is a sequence of complex numbers. The images of $1, 2, \dots, n, \dots$ under $f$ are usually denoted by $z_1, z_2, \dots, z_n, \dots$ and which are called first, second, $\dots$ , $n$ -th term, $\dots$ respectively of sequence. This sequence is denoted by symbols ${z_n}$ or $\{z_1, z_2, \dots, z_n \}$ or $\langle z_n \rangle$ .
(11.) Limit of a sequence
A sequence ${z_n}$ is said to have a limit $z_0$ , written $\underset{n \to \infty}{\lim} z_n = z_0$ , if for every $\epsilon > 0$ , there exists a positive integer $n_0(\epsilon)$ such that $|z_n - z_0| < \epsilon$ whenever $n > n_0(\epsilon)$ . Geometrically, this means that every neighbourhood of $z_0$ contains all but a finite number of terms of the sequence.
(12.) Convergent sequence
If $\underset{n \to \infty}{\lim} z_n = z_0$ is a finite quantity, then complex sequence $\{z_n\}$ is said to be convergent sequence. In this case we say that the sequence $\{z_n \}$ converges to $z_0$ . There is a nice relationship between the convergence of a sequence of complex numbers and the convergence of its real and imaginary parts.
Theorem: 2. Let ${z_n} = {x_n + i y_n}$ be a sequence of complex numbers. Then $\{z_n\}$ converges to a complex number $z_0 = x_0 + i y_0$ if and only if ${x_n}$ converges to $x_0$ and ${y_n}$ converges to $y_0$ .
Proof. Assume that $\underset{n \to \infty}{\lim} x_n = x_0$ and $\underset{n \to \infty}{\lim} y_n = y_0$ . Then given $\epsilon > 0$ , there exists an integer $n_0(\epsilon)$ such that $n > n_0(\epsilon)$ implies
$|x_n - x_0| < \frac{\epsilon}{2}, \quad |y_n - y_0| < \frac{\epsilon}{2}$
Then, for $n > n_0(\epsilon)$
$|z_n - z_0| = |(x_n - x_0) + i(y_n - y_0)| \le |x_n - x_0| + |y_n - y_0| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$
and thus ${z_n}$ converges to $z_0$ . Conversely, if we assume that $\underset{n \to \infty}{\lim} z_n = z_0$ , then the inequalities $|x_n - x_0| \le |z_n - z_0|$ , $|y_n - y_0| \le |z_n - z_0|$ show that $\{x_n\}$ and $\{y_n\}$ converge to $x_0$ and $y_0$ respectively.
Thus above theorem shows that many properties of complex sequences may be deduced from corresponding properties of real sequences. Now we have following theorems which are similar to theorems for real sequences.
Theorem: 3. A convergent sequence $\{z_n\}$ is bounded.
Proof: If $\underset{n \to \infty}{\lim} z_n = z_0$ , then $z_n \in N_\epsilon(z_0)$ for all $n > n_0(\epsilon)$ .
Let $M = \max{|z_1|, |z_2|, \dots, |z_{n_0}|}$ . Then $|z_n| < M$ for every $n$ and hence $\{z_n\}$ is bounded.
A subsequence of a sequence $\{z_n\}$ is a sequence ${z_{n_k}}$ whose terms are selected from the terms of the original sequence and arranged in the same order.
Theorem: 4. If a sequence $\{z_n\}$ converges to $z_0$ , then every subsequence ${z_{n_k}}$ converges to $z_0$ .
Proof: Given $\epsilon > 0$ , we have $z_n \in N_\epsilon(z_0)$ for all $n > n_0(\epsilon)$ . Hence $z_{n_k} \in N_\epsilon(z_0)$ for $n_k > n_0(\epsilon)$ . Since $n_k \ge n_0(\epsilon)$ , there can be at the most $n_0(\epsilon)$ terms of the subsequence for which $|z_{n_k} - z_0| \ge \epsilon$ . Hence ${z_{n_k}}$ converges to $z_0$ .
Theorem 5. A point $z_0$ is a limit point of a set $A \subset \mathbb{C}$ if and only if there is a sequence of distinct points in A converging to $z_0$ .
Proof: If a sequence $\{z_n\}$ of distinct point in A converges to $z_0$ , then every neighbourhood of $z_0$ contains all but a finite number (hence infinitely many) points of $\{z_n\}$ . Therefore, $z_0$ must be a limit point of A.
Conversely, let $z_0$ be a limit point of A. For every integer $n$ , choose a point $z_n \in N_{\frac{1}{n}}(z_0) \cap A$ . Since every neighbourhood of A contains infinitely many distinct points, we may assume the points of sequence $\{z_n\}$ to be distinct.
Given $\epsilon > 0$ , choose $n_0(\epsilon)$ such that $\frac{1}{n_0} < \epsilon$ . Then $z_n \in N_\epsilon(z_0)$ for all $n > n_0(\epsilon)$ and thus the sequence $\{z_n\}$ converges to $z_0$ .
Analogous to real line we have the following result.
Theorem: 6. Bolzano-Weierstrass: Every bounded infinite set in the complex plane has a limit point.
Proof: Choose any sequence $\{z_n\}$ of distinct points in the set. This sequence contains a convergent sub-sequence ${z_{n_k}}$ ; and the limit of this convergent sub-sequence is a limit point of the set.
Definition: (Cauchy sequence): A sequence $\{z_n\}$ of complex numbers is said to be a Cauchy sequence if for every $\epsilon > 0$ there exists an integer $n_0(\epsilon)$ such that $|z_m - z_n| < \epsilon$ whenever $m, n > n_0(\epsilon)$ .
In a convergent sequence, all but a finite number of points are close to a fixed point (the limit of the sequence), while for Cauchy sequence all but a finite number of points are close to each other. In the next theorem we will show that for complex sequences these two concepts are equivalent.
By understanding the above illutrations,one can understand the Topological Properties .

📢 If you liked this math article: * 👥 **Share with Friends:** Knowledge grows by sharing, so be sure to share it with your friends. * 🔔 **Follow the Website:** If this is your first time here, follow our **email subscription** so that you get instant notifications for every new article. * 💬 **Give Your Suggestions:** If you have any issues or would like to make any suggestions, do let us know by **commenting** below. *Welcome to read the full article!*

Also Read This Article:- Geometrical Construction of Complex Numbers on Argand Plane

3.Frequently Asked Questions Related to Topological Properties of Complex Numbers

Q:1.What is a point set?

Ans:Any collection of points in the complex plane is called a point set and each point is called a member or element the set. Now we give the following fundamental definitions which will required to study a point set.

Q: 2. What do you mean by neighbourhood?

Ans: A delta or $\delta$ -neighbourhood of a point $z_0$ is the set of all points $z$ such that $|z - z_0| < \delta$ .
where $\delta$ is any given positive number. For nbd. of a point $z_0$ we use the notation $N_\delta(z_0)$ thus
$N_\delta(z_0) = \{z \in \mathbb{C} : |z - z_0| < \delta \}$
If $z_0$ is deleted from the neighbourhood of the point $z_0$ then it is called deleted $\delta$ -nbd which is denoted as $N'_\delta(z_0)$ .
Thus
$N'_\delta(z_0) = \{z \in \mathbb{C} : 0 < |z - z_0| < \delta \}$
Obviously $\delta$ -nbd of $z_0$ is the whole region lying inside a circle in Argand diagram with centre at $z_0$ and radius $\delta$ i.e. an open disk in complex plane with centre at $z_0$ and radius $\delta$ .

Q: 3. Define the Limit Point

Ans: A point $z_0$ is called a limit point, cluster point, or point of accumulation of a point set S if every deleted nbd of $z_0$ contains at least one point of S. i.e. if $N'_\delta(z_0) \cap S \neq \phi$ . Since $\delta$ can be any positive number it follows that if $z_0$ is a limit point of S, then every nbd of $z_0$ contains infinitely many points of S. Note that $z_0$ may or may not belong to the set S.
By answering the above questions,you can know about the primary terms of Topological Properties .

Also Read This Article:- Stereographic Projection and Inverse Points of Complex Numbers

4.छात्र-छात्राओं से आज का प्रश्न (Today’s Question to Students):

Topological Properties of Complex Numbers

🎯 विनर्स कॉर्नर:क्या आप इस सवाल का सही जवाब दे सकते हैं?अपने नाम के साथ नीचे कमेंट करें!सही जवाब देने वाले Top Students के नाम हमारी आगे post/is article ke update में Photo ya Special Mention के साथ publish की जाएगी।अपना जवाब अभी दर्ज करें!👇
*”कोई संख्या अपने सप्तांश से 18 अधिक है,वह संख्या क्या है?”*
(A number is 18 more than its seventh, what is that number?)
*पिछली प्रश्नोत्तरी का हल*
$\frac{72}{8} - \frac{48}{6} = 9 - 8 = 1$
*Previous Quiz Solution*
$\frac{72}{8} - \frac{48}{6} = 9 - 8 = 1$
*”This article has been prepared by **Satyam Coaching Centre** on the **Satyam Mathematics** blog.”*

About Author

Sanjay Kumawat

(1.)**Satyam Narain Kumawat** **Website Name:Satyam Mathematics** *Owner:satyamcoachingcentre.in* *Sthan:Manoharpur,Jaipur (Rajasthan)* **Teaching Mathematics aur Anya Anubhav** ***Shiksha:**B.sc.,B.Ed.,(M.sc. star Ke Mathematics Ko Padhane ka Anubhav),B.com.,M.com. Ke vishayon Ko Padhane ka Anubhav,Philosophy,Psychology,Religious,sanskriti Mein Gahri Ruchi aur Adhyayan ***Anubhav:**phichale 23 varshon se M.sc.,M.com.,Angreji aur Vigyan Vishayon Mein Shikshaka Ka Lamba Anubhav ***Visheshagyata:*Maths,Adhyatma (spiritual),Yog vishayon ka vistrit Gyan* ****In Brief:I have read about M.sc. books,psychology,philosophy,spiritual, vedic,religious,yoga,health and different many knowledgeable books.A dedicated math expert with 23+ years of teaching experience upto M.sc. ,M.com.,English and science.After guiding thousands of students through Satyam Coaching Center,now share Mathematics,Trigonometry (Upto M.sc) and Educational Strategies in simple language on this blog from December 2018.* (2.)**(Technical Expert & Co-Admin):** ***Name:Sanjay Kumawat* *Qualification:Graduate in Mechanical Engineering (B.Tec) in 2013* *Profession:Physics Lecturer* *Teaching Experience:15 Years and Teaching to NEET,JEE Students* *Technical Experience:5 Years Coding and Article Editing,Classic Photo Editing by Laptop in Satyam Coaching Centre Blog* *A school lecturer and digital content strategist.On this blog,he handles all the responsibility of coding,image editing,SEO, and technical management,so that the mathematical content reaches the readers in a very accurate and beautiful form.* Updated on 15.06.2026