Real Analysis

Neighbourhood

A neighbourhood of a point in a metric space is a set containing an open ball centered at that point, serving as the foundational topological tool for defining limits, continuity, and interior points. This topic covers neighbourhood properties and punctured neighbourhoods.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Definition of Neighbourhood and Open Intervals

Let $(X, d)$ be a metric space, $x_0 \in X$, and $r > 0$. The open ball $B(x_0, r) = \{x \in X : d(x, x_0) < r\}$ is called a neighbourhood of $x_0$. In the standard metric space $(\mathbb{R}, |\cdot|)$, an $\epsilon$-neighbourhood of a point $x_0 \in \mathbb{R}$ is the open interval $(x_0 - \epsilon, x_0 + \epsilon)$ for any $\epsilon > 0$.

📐 Geometric Interpretation

Geometrically, a neighbourhood represents a symmetric open span or circular/spherical zone around a target point $x_0$ with radius $\epsilon$, completely excluding the boundary points when strictly defined as an open set.

⚙️ Algebraic & Structural Meaning

Algebraically, it translates inequalities of the form $|x - x_0| < \epsilon$ into structural set-theoretic properties, enabling the rigorous formulation of limits and topological interiors.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Positivity of Radius	`\epsilon > 0`	Strictly Mandatory / Necessary	If $\epsilon = 0$, the neighbourhood collapses to a singleton set $\{x_0\}$, destroying the open set property required for limit operations.
Metric Space Foundation	`(X, d) metric space`	Strictly Mandatory	Without a distance function satisfying non-negativity, symmetry, and triangle inequality, the concept of 'nearness' becomes undefined.
Containment of Open Ball	`N \supset B(x_0, \epsilon)`	Essential for general topology	If a set $N$ contains $x_0$ but contains no open ball around $x_0$, $N$ fails to be a neighbourhood of $x_0$ (e.g., singleton sets in $\mathbb{R}$).

Useful results and shortcuts

Use these identities and results when their conditions apply:

Singleton and Finite Set Neighbourhoods

$$ \text{Int}(\text{Finite Set}) = \emptyset $$

Rule: No finite set in $\mathbb{R}$ can be a neighbourhood of any of its points because it contains no open interval.

Application: Use this to trap candidates on whether integers $\mathbb{Z}$ or finite sets are neighbourhoods.

Set Inclusion Test for Neighbourhood

$$ x_0 \in (a, b) \implies \exists \epsilon > 0 \text{ s.t. } (x_0-\epsilon, x_0+\epsilon) \subset (a, b) $$

Rule: To check if a set $S$ is a neighbourhood of $x_0$, verify if $x_0$ lies strictly in the interior of $S$.

Application: Directly applied in DSSSB and KVS questions testing interior points of closed vs open intervals.

Countable vs Uncountable Neighbourhood Intersection

$$ \bigcap_{n=1}^{\infty} \left(-\frac{1}{n}, \frac{1}{n}\right) = \{0\} $$

Rule: Arbitrary or infinite intersection of open neighbourhoods may reduce to a non-open set (a singleton).

Rational and Irrational Neighbourhood Density

$$ \text{Int}(\mathbb{Q}) = \emptyset, \quad \text{Int}(\mathbb{R} \setminus \mathbb{Q}) = \emptyset $$

Rule: Neither the set of rationals nor irrationals contains any open interval, hence neither is a neighbourhood of any real number.

Application: Classic analytical distinction tested in PGT Mathematics.

Punctured Neighbourhood Radius Bound

$$ N^*(x_0, \delta) = B(x_0, \delta) \setminus \{x_0\} $$

Rule: When evaluating limits $\lim_{x \to x_0} f(x)$, the function need not be defined at $x_0$, relying entirely on the punctured neighbourhood.

Application: Crucial for limit existence questions in calculus and real analysis.

Rapid Elimination via Open Intervals

$$ S \text{ is nbd of } x \iff x \in \text{Int}(S) $$

Rule: Eliminate any option claiming a closed boundary point or discrete point forms a neighbourhood in standard $\mathbb{R}$.

Application: Saves up to 45 seconds in objective screening tests.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Closed Interval Endpoint Neighbourhood Fallacy

Common Pitfall: Candidates assume that a closed interval $[a, b]$ is a neighbourhood of its endpoint $a$ or $b$.

Counterexample: [0, 1] is not a neighbourhood of 0 in \mathbb{R}

Why it fails: Any open interval around $0$, say $(-\epsilon, \epsilon)$, contains negative numbers outside $[0, 1]$, violating the requirement that the neighbourhood must be a subset of the set.

🛡️ Exam Defense: A set can only be a neighbourhood of points lying strictly in its interior.

⚠️ COMMON ERROR #02

Natural Numbers and Integers

Common Pitfall: Assuming discrete sets like $\mathbb{N}$ or $\mathbb{Z}$ contain neighbourhoods for their elements.

Counterexample: \mathbb{Z} \text{ in } \mathbb{R}

Why it fails: No open interval around an integer $n$ is entirely contained within $\mathbb{Z}$ due to the presence of fractions.

🛡️ Exam Defense: Discrete sets have empty interiors in the standard topology of $\mathbb{R}$.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Open Neighbourhood	\epsilon > 0, x_0 \in \mathbb{R}	(x_0 - \epsilon, x_0 + \epsilon)	Symmetric open interval around $x_0$	Appears in limit and continuity definitions
Punctured Neighbourhood	0 < \|x - x_0\| < \delta	B(x_0, \delta) \setminus \{x_0\}	Open interval excluding the center point	Key for evaluating limits where $f(x_0)$ may fail
Closed Interval [a, b]	a < b in \mathbb{R}	Contains boundary points $a$ and $b$	Interval including endpoints	Not a nbd of its endpoints; nbd of interior points
Set of Rational Numbers \mathbb{Q}	Dense subset of \mathbb{R}	\text{Int}(\mathbb{Q}) = \emptyset	Scattered without open intervals	Never a neighbourhood of any real number

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Which of the following sets is a neighbourhood of the point $x = 2$ in the usual topology of $\mathbb{R}$?

(A) [1, 2](B) (1.5, 2.5](C) (1.9, 2.1)(D) \{2, 2.1, 2.2\}

📚 Method A: Formal Textbook Derivation

Step 1: A set $S$ is a neighbourhood of $2$ if there exists an open interval $(2 - \epsilon, 2 + \epsilon)$ contained in $S$. Step 2: Test option (C): $(1.9, 2.1)$ is an open interval containing $2$ and is entirely within itself. Step 3: Options (A) and (B) fail because $2$ is a right/left endpoint where no open radius can fit entirely inside the set without spilling outside. Step 4: Option (D) is finite, hence has empty interior.

⚡ Method B: 25-Second Exam Shortcut

Instantly check for strict open interval containment around the point. Only (C) is an open set containing 2.

💡 Tip: Look for open brackets around the target point for standard neighbourhood questions on $\mathbb{R}$.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Let $S = \left\{ \frac{1}{n} : n \in \mathbb{N} \right\}$. Is $S$ a neighbourhood of any point in $\mathbb{R}$?

(A) Yes, of $0$(B) Yes, of every point in $S$(C) No, because $S$ has no interior points(D) Yes, of $1$

📚 Method A: Formal Textbook Derivation

Step 1: Examine the set $S = \{1, 1/2, 1/3, \dots\}$. Step 2: For any element in $S$ or any real number, try to construct an open interval around it. Step 3: Any open interval around any point in $\mathbb{R}$ contains infinitely many irrational and rational numbers not in $S$. Step 4: Thus, $\text{Int}(S) = \emptyset$, meaning $S$ is not a neighbourhood of any point.

⚡ Method B: 25-Second Exam Shortcut

Countable sets consisting of isolated points in $\mathbb{R}$ possess zero interior and cannot form neighbourhoods.

💡 Tip: Remember that isolated points or sequences converging to zero never form neighbourhoods in the standard metric space.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: Which of the following intervals is a neighbourhood of $x = 3$?

A [2, 3]

B (2.5, 3.5)

C [3, 4)

D \{3\}

Derivation & Analysis:

An interval must be open and contain $3$ in its interior to be a neighbourhood of $3$. Only $(2.5, 3.5)$ satisfies this.

MCQ 2: The interior of the set of integers $\mathbb{Z}$ in $(\mathbb{R}, |\cdot|)$ is:

A $\mathbb{Z}$

B $\mathbb{R}$

C $\emptyset$

D $\mathbb{Q}$

Derivation & Analysis:

Integers are isolated points in $\mathbb{R}$; no open interval can be found within $\mathbb{Z}$. Thus, $\text{Int}(\mathbb{Z}) = \emptyset$.

MCQ 3: A punctured neighbourhood of a point $x_0$ with radius $\delta > 0$ is denoted by:

A $(x_0 - \delta, x_0 + \delta)$

B $[x_0 - \delta, x_0 + \delta]$

C $(x_0 - \delta, x_0 + \delta) \setminus \{x_0\}$

D $\{x_0\}$

Derivation & Analysis:

By definition, a punctured neighbourhood removes the center point $x_0$ from the open ball.

MCQ 4: If $A$ and $B$ are neighbourhoods of $x_0$, then $A \cap B$ is:

A Not necessarily a neighbourhood

B Always a neighbourhood of $x_0$

C An empty set

D A closed set always

Derivation & Analysis:

The finite intersection of neighbourhoods is a neighbourhood because the intersection of two open balls contains a smaller open ball centered at $x_0$.

MCQ 5: Which of the following statements is true regarding open intervals in $\mathbb{R}$?

A Every open interval is a neighbourhood of each of its points.

B No open interval is a neighbourhood.

C Open intervals are neighbourhoods only of their endpoints.

D Open intervals are closed sets.

Derivation & Analysis:

By definition, for any point in an open interval $(a, b)$, an $\epsilon$-neighbourhood can be constructed entirely within $(a, b)$.

MCQ 6: The set of rational numbers $\mathbb{Q}$ is a neighbourhood of:

A Every rational number

B Zero

C No point in $\mathbb{R}$

D Every real number

Derivation & Analysis:

Since $\text{Int}(\mathbb{Q}) = \emptyset$, $\mathbb{Q}$ contains no open intervals and is not a neighbourhood of any point.

MCQ 7: If $N$ is a neighbourhood of $x_0$ and $N \subset M$, then $M$ is:

A Not a neighbourhood of $x_0$

B Always a neighbourhood of $x_0$

C A closed set

D A bounded set

Derivation & Analysis:

Superset of a neighbourhood is also a neighbourhood because any open ball contained in $N$ is also contained in $M$.

MCQ 8: Assertion (A): Every neighbourhood of a point $x_0 \in \mathbb{R}$ contains infinitely many points. Reason (R): Every neighbourhood contains an open interval of positive length.

A Both (A) and (R) are true and (R) is the correct explanation of (A).

B Both (A) and (R) are true but (R) is not the correct explanation.

C (A) is true but (R) is false.

D (A) is false but (R) is true.

Derivation & Analysis:

An open interval of positive length $\epsilon > 0$ always contains uncountably many real numbers, making both assertion and reason true.

MCQ 9: Let $S = (0, 1] \cup \{2\}$. $S$ is a neighbourhood of:

A All points in $(0, 1)$

B The point $2$

C The point $1$

D No point

Derivation & Analysis:

Only points in the open interval $(0, 1)$ have an open ball around them entirely contained in $S$. Points $1$ and $2$ fail.

Q10

MCQ 10: In a discrete metric space $(X, d)$ where $d(x, y) = 1$ for $x \neq y$, the singleton set $\{x_0\}$ is:

A Not a neighbourhood of $x_0$

B A neighbourhood of $x_0$

C An infinite set

D Never an open set

Derivation & Analysis:

In a discrete metric space, the open ball $B(x_0, 0.5) = \{x_0\}$, making every singleton set an open set and thus a neighbourhood of its point.

Common questions

A closed interval $[a, b]$ is a neighbourhood of all its interior points $(a, b)$, but it is NOT a neighbourhood of its endpoints $a$ and $b$ in the standard topology of $\mathbb{R}$. This is a frequent trap in UP PGT exams.

An open set is a neighbourhood of each of its points. However, a neighbourhood of a point $x_0$ does not need to be an open set overall; it only needs to contain some open set (or open ball) that surrounds $x_0$.

Instantly check whether you can draw an open interval around the point that lies entirely inside the set. If the point touches the boundary or is isolated, reject the option immediately.