The Central Idea

In ordinary real analysis, the distance between two real numbers is

\[ |x-y|. \]

Metric-space theory asks which parts of analysis depend only on the properties of distance rather than on the special algebraic structure of the real numbers.

The answer is extensive. If a set has a suitable notion of distance, then we can already discuss limits, continuity, neighborhoods, closedness, completeness, and compactness.

What Is a Metric Space?

A metric space is a pair

\[ (X,d), \]

where \(X\) is a set and

\[ d:X\times X\to[0,\infty) \]

is a function called a metric.

For two points \(x,y\in X\), the value

\[ d(x,y) \]

represents the distance from \(x\) to \(y\).

The Four Metric Axioms

A function \(d\) is a metric if, for all \(x,y,z\in X\), it satisfies four requirements.

  1. Nonnegativity: \(d(x,y)\ge0\).
  2. Identity of indiscernibles: \(d(x,y)=0\) if and only if \(x=y\).
  3. Symmetry: \(d(x,y)=d(y,x)\).
  4. Triangle inequality: \(d(x,z)\le d(x,y)+d(y,z)\).

These axioms capture the structural behavior expected from distance while allowing the underlying objects to be extremely general.

Why the Triangle Inequality Matters

The triangle inequality says that traveling from \(x\) to \(z\) directly cannot be longer than traveling from \(x\) to \(z\) through an intermediate point \(y\):

\[ \boxed{d(x,z)\le d(x,y)+d(y,z)}. \]

This inequality is one of the main engines of analysis. It lets us compare errors, prove uniqueness of limits, control approximations, and combine several local estimates into one global estimate.

The Usual Metric on the Real Numbers

On \(\mathbb{R}\), the standard metric is

\[ d(x,y)=|x-y|. \]

For example,

\[ d(2,7)=|2-7|=5. \]

The usual definitions of limits and continuity on the real line can be rewritten entirely in terms of this metric.

The Euclidean Metric on \(\mathbb{R}^n\)

For points

\[ x=(x_1,\ldots,x_n), \qquad y=(y_1,\ldots,y_n), \]

the Euclidean metric is

\[ \boxed{ d_2(x,y)= \sqrt{\sum_{k=1}^{n}(x_k-y_k)^2} }. \]

In \(\mathbb{R}^2\), this is the familiar distance formula from coordinate geometry.

Different Metrics on the Same Set

The same underlying set may carry more than one metric.

On \(\mathbb{R}^n\), another common metric is the taxicab metric

\[ d_1(x,y)=\sum_{k=1}^{n}|x_k-y_k|. \]

A third is the maximum metric

\[ d_\infty(x,y)=\max_{1\le k\le n}|x_k-y_k|. \]

These metrics assign different numerical distances, but on finite-dimensional Euclidean space they generate the same notions of convergence and open sets.

Example: Comparing Metrics in \(\mathbb{R}^2\)

Let

\[ x=(1,2), \qquad y=(4,6). \]

Then

\[ d_2(x,y)=\sqrt{3^2+4^2}=5, \] \[ d_1(x,y)=3+4=7, \]

and

\[ d_\infty(x,y)=4. \]

The values differ because the metrics represent different rules for measuring separation.

The Discrete Metric

Every set \(X\) can be turned into a metric space using the discrete metric

\[ d(x,y)= \begin{cases} 0, & x=y,\\ 1, & x\ne y. \end{cases} \]

Under this metric, distinct points are always exactly one unit apart.

This example is important because it shows that metric-space theory does not require geometry in the usual visual sense.

Metrics on Spaces of Functions

The points of a metric space do not have to be numbers or coordinate vectors. They can be functions.

For continuous functions on \([a,b]\), define

\[ d_\infty(f,g)= \max_{x\in[a,b]}|f(x)-g(x)|. \]

This measures the largest vertical separation between the two functions. It is the metric associated with uniform convergence.

An Integral Metric on Functions

On suitable classes of functions, another distance can be defined by

\[ d_1(f,g)= \int_a^b |f(x)-g(x)|\,dx. \]

Rather than measuring the largest pointwise error, this metric measures total accumulated separation.

Different function-space metrics therefore encode different meanings of approximation.

Open Balls

Let \((X,d)\) be a metric space. The open ball centered at \(x\in X\) with radius \(r>0\) is

\[ \boxed{ B_r(x)=\{y\in X:d(x,y)<r\} }. \]

An open ball collects all points whose distance from the center is strictly less than the radius.

Closed Balls

The corresponding closed ball is

\[ \overline{B}_r(x)= \{y\in X:d(x,y)\le r\}. \]

The distinction is the boundary condition:

\[ d(x,y)<r \]

for an open ball, and

\[ d(x,y)\le r \]

for a closed ball.

The Shape of a Ball Depends on the Metric

In \(\mathbb{R}^2\), a ball centered at the origin looks different under different metrics.

The algebraic definition of the metric determines the geometry of the space.

Neighborhoods

A set \(N\subseteq X\) is a neighborhood of a point \(x\in X\) if it contains some open ball centered at \(x\).

In other words, there exists \(r>0\) such that

\[ B_r(x)\subseteq N. \]

Neighborhoods provide a precise way to describe what happens locally near a point.

Open Sets

A subset \(U\subseteq X\) is open if every point of \(U\) has some open ball lying completely inside \(U\).

Formally,

\[ \boxed{ x\in U \Longrightarrow \exists r>0\text{ such that }B_r(x)\subseteq U }. \]

Openness means that every point has some room around it before leaving the set.

Example: An Open Interval

In \(\mathbb{R}\) with the usual metric, the interval

\[ (2,5) \]

is open.

If \(x\in(2,5)\), choose

\[ r=\frac12\min\{x-2,5-x\}. \]

Then

\[ B_r(x)\subseteq(2,5). \]

Every point therefore has a sufficiently small neighborhood contained in the interval.

Closed Sets

A subset \(F\subseteq X\) is closed if its complement

\[ X\setminus F \]

is open.

In metric spaces, an equivalent characterization is especially useful: \(F\) is closed if every convergent sequence of points in \(F\) has its limit in \(F\).

Open and Closed Are Not Opposites

A set can be open, closed, both, or neither.

In every metric space, the entire space \(X\) and the empty set \(\varnothing\) are both open and closed.

On \(\mathbb{R}\), the half-open interval

\[ [0,1) \]

is neither open nor closed in the usual metric.

Interior Points

A point \(x\in A\) is an interior point of a set \(A\subseteq X\) if some open ball around \(x\) lies entirely in \(A\).

The set of all interior points of \(A\) is called the interior of \(A\), written

\[ A^\circ. \]

A set is open exactly when

\[ A=A^\circ. \]

Limit Points

A point \(x\in X\) is a limit point or accumulation point of \(A\subseteq X\) if every open ball around \(x\) contains a point of \(A\) different from \(x\).

Symbolically, for every \(r>0\),

\[ \bigl(B_r(x)\setminus\{x\}\bigr)\cap A\ne\varnothing. \]

A closed set contains all of its limit points.

Closure

The closure of \(A\), written

\[ \overline{A}, \]

is the smallest closed set containing \(A\).

In a metric space, \(\overline{A}\) consists of the points of \(A\) together with all limit points of \(A\).

Equivalently,

\[ x\in\overline{A} \quad\Longleftrightarrow\quad B_r(x)\cap A\ne\varnothing \text{ for every }r>0. \]

Boundary Points

A point \(x\in X\) is a boundary point of \(A\) if every open ball centered at \(x\) meets both \(A\) and its complement.

The boundary of \(A\) is written

\[ \partial A. \]

One useful identity is

\[ \boxed{ \partial A=\overline{A}\setminus A^\circ }. \]

Convergence in a Metric Space

A sequence \((x_n)\) in \((X,d)\) converges to \(x\in X\) if for every \(\varepsilon>0\), there exists \(N\in\mathbb{N}\) such that

\[ n\ge N \quad\Longrightarrow\quad d(x_n,x)<\varepsilon. \]

We write

\[ \boxed{x_n\to x}. \]

This is the usual \(\varepsilon\)-definition of sequence convergence with absolute value replaced by a general metric.

Example: Coordinatewise Convergence in \(\mathbb{R}^n\)

Under the usual Euclidean metric, a sequence

\[ x_n=(x_{n,1},\ldots,x_{n,k}) \]

converges to

\[ x=(x_1,\ldots,x_k) \]

exactly when each coordinate converges:

\[ x_{n,j}\to x_j \qquad \text{for every }j. \]

Limits Are Unique

A sequence in a metric space cannot converge to two different points.

Suppose

\[ x_n\to x \qquad\text{and}\qquad x_n\to y. \]

By the triangle inequality,

\[ d(x,y) \le d(x,x_n)+d(x_n,y). \]

Both terms on the right can be made arbitrarily small, forcing

\[ d(x,y)=0. \]

Therefore,

\[ \boxed{x=y}. \]

Bounded Sets

A subset \(A\subseteq X\) is bounded if there exist a point \(x_0\in X\) and a radius \(R>0\) such that

\[ A\subseteq B_R(x_0). \]

Equivalently, all points of \(A\) lie within some finite distance of one fixed center.

The Diameter of a Set

The diameter of a nonempty set \(A\subseteq X\) is

\[ \operatorname{diam}(A) =\sup\{d(x,y):x,y\in A\}. \]

A set is bounded exactly when its diameter is finite.

Diameter measures the largest possible internal separation within the set.

Cauchy Sequences

A sequence \((x_n)\) in \((X,d)\) is Cauchy if its terms eventually become arbitrarily close to one another.

Formally, for every \(\varepsilon>0\), there exists \(N\) such that

\[ m,n\ge N \quad\Longrightarrow\quad d(x_m,x_n)<\varepsilon. \]

The definition does not name a candidate limit. It asks whether the tail of the sequence is internally coherent.

Every Convergent Sequence Is Cauchy

Suppose

\[ x_n\to x. \]

Then for sufficiently large \(m\) and \(n\), both \(x_m\) and \(x_n\) are close to \(x\). By the triangle inequality,

\[ d(x_m,x_n) \le d(x_m,x)+d(x,x_n). \]

Each term on the right can be made smaller than \(\varepsilon/2\), so

\[ d(x_m,x_n)<\varepsilon. \]

Thus convergence always implies the Cauchy property.

Complete Metric Spaces

A metric space \((X,d)\) is complete if every Cauchy sequence in \(X\) converges to a point that also belongs to \(X\).

\[ \boxed{ \text{Cauchy in }X \Longrightarrow \text{convergent in }X }. \]

Completeness is therefore a statement that the space contains all limits demanded by its own internal approximation processes.

Example: \(\mathbb{R}\) Is Complete

With the usual metric

\[ d(x,y)=|x-y|, \]

the real numbers form a complete metric space.

This is the metric-space form of real-number completeness: every real Cauchy sequence converges to a real number.

Example: \(\mathbb{Q}\) Is Not Complete

The rational numbers with the usual metric are not complete.

There are rational sequences that approximate \(\sqrt2\) arbitrarily well. Such a sequence is Cauchy in \(\mathbb{Q}\), but its natural limit is not rational.

Thus the sequence has no limit inside the metric space \(\mathbb{Q}\).

Closed Subsets of Complete Spaces

If \((X,d)\) is complete and \(F\subseteq X\) is closed, then \(F\) is complete under the metric inherited from \(X\).

The reason is structural: a Cauchy sequence in \(F\) converges somewhere in \(X\) because \(X\) is complete, and the limit lies in \(F\) because \(F\) is closed.

Completeness Forces Closedness Inside a Complete Ambient Space

Conversely, if \(A\subseteq X\) is complete under the inherited metric and \(X\) is a metric space containing \(A\), then \(A\) must be closed in \(X\).

Thus within a complete metric space,

\[ \boxed{ A\text{ is complete} \Longleftrightarrow A\text{ is closed} } \]

for subsets equipped with the inherited metric.

Continuity Between Metric Spaces

Let

\[ f:(X,d_X)\to(Y,d_Y). \]

The function \(f\) is continuous at \(x\in X\) if for every \(\varepsilon>0\), there exists \(\delta>0\) such that

\[ d_X(x,y)<\delta \quad\Longrightarrow\quad d_Y(f(x),f(y))<\varepsilon. \]

This is the familiar \(\varepsilon\)-\(\delta\) definition expressed using the metrics of the domain and codomain.

Sequential Characterization of Continuity

In metric spaces, continuity can be tested using sequences.

A function \(f:X\to Y\) is continuous at \(x\) exactly when

\[ x_n\to x \quad\Longrightarrow\quad f(x_n)\to f(x). \]

This equivalence is one reason sequences are so useful in metric-space analysis.

Continuity and Open Sets

A function

\[ f:X\to Y \]

is continuous on \(X\) if and only if the inverse image of every open set in \(Y\) is open in \(X\).

\[ \boxed{ V\subseteq Y\text{ open} \Longrightarrow f^{-1}(V)\subseteq X\text{ open} }. \]

This formulation becomes fundamental when analysis moves toward topology.

Lipschitz Continuity

A function

\[ f:(X,d_X)\to(Y,d_Y) \]

is Lipschitz continuous if there exists \(L\ge0\) such that

\[ d_Y(f(x),f(y)) \le Ld_X(x,y) \]

for all \(x,y\in X\).

Every Lipschitz function is uniformly continuous.

Uniform Continuity

A function \(f:X\to Y\) is uniformly continuous if for every \(\varepsilon>0\), there exists \(\delta>0\) such that

\[ d_X(x,y)<\delta \quad\Longrightarrow\quad d_Y(f(x),f(y))<\varepsilon \]

for all \(x,y\in X\).

The important point is that the same \(\delta\) works everywhere in the domain.

Compactness by Open Covers

A subset \(K\subseteq X\) is compact if every open cover of \(K\) has a finite subcover.

If

\[ K\subseteq\bigcup_{\alpha\in A}U_\alpha \]

with each \(U_\alpha\) open, compactness guarantees that finitely many of those sets already cover \(K\):

\[ K\subseteq U_{\alpha_1}\cup\cdots\cup U_{\alpha_m}. \]

Sequential Compactness

In metric spaces, compactness is equivalent to a sequential property: every sequence in \(K\) has a subsequence that converges to a point of \(K\).

\[ \boxed{ K\text{ compact} \Longleftrightarrow \text{every sequence in }K\text{ has a convergent subsequence with limit in }K }. \]

This equivalence connects the global open-cover definition with the local behavior of sequences.

Compact Sets Are Closed and Bounded

Every compact subset of a metric space is closed and bounded.

The converse is not true in every metric space.

The familiar equivalence

\[ \boxed{ K\subseteq\mathbb{R}^n\text{ compact} \Longleftrightarrow K\text{ closed and bounded} } \]

is the Heine–Borel theorem and depends on the special structure of finite-dimensional Euclidean space.

Compactness and Continuous Functions

Continuous images of compact sets are compact.

Therefore, if

\[ f:K\to Y \]

is continuous and \(K\) is compact, then

\[ f(K) \]

is compact in \(Y\).

Many familiar extreme-value and boundedness results are consequences of this principle.

Continuous Functions on Compact Metric Spaces

A continuous real-valued function on a compact metric space is bounded and attains both its maximum and minimum values.

Moreover, every continuous function from a compact metric space into a metric space is uniformly continuous.

Compactness therefore upgrades local continuity into stronger global control.

Totally Bounded Sets

A metric space \(X\) is totally bounded if for every \(\varepsilon>0\), the space can be covered by finitely many open balls of radius \(\varepsilon\).

Thus there exist points \(x_1,\ldots,x_m\in X\) such that

\[ X\subseteq B_\varepsilon(x_1)\cup\cdots\cup B_\varepsilon(x_m). \]

Total boundedness is stronger than ordinary boundedness.

Compactness as Completeness Plus Total Boundedness

A fundamental characterization in metric spaces is

\[ \boxed{ X\text{ compact} \Longleftrightarrow X\text{ complete and totally bounded} }. \]

This result separates compactness into two ideas: no Cauchy limit is missing, and the space can be approximated at every scale by finitely many points.

Distance from a Point to a Set

If \(A\subseteq X\) is nonempty, define the distance from \(x\in X\) to \(A\) by

\[ d(x,A)=\inf_{a\in A}d(x,a). \]

This quantity measures how close \(x\) can get to points of \(A\).

If \(A\) is closed, then

\[ d(x,A)=0 \quad\Longleftrightarrow\quad x\in A. \]

Dense Subsets

A subset \(D\subseteq X\) is dense in \(X\) if

\[ \overline{D}=X. \]

Equivalently, every point of \(X\) can be approximated arbitrarily closely by points of \(D\).

The rational numbers are dense in the real numbers:

\[ \overline{\mathbb{Q}}=\mathbb{R}. \]

Separable Metric Spaces

A metric space is separable if it contains a countable dense subset.

Since \(\mathbb{Q}\) is countable and dense in \(\mathbb{R}\), the real line is separable.

Separable spaces can often be approximated using only countably many reference points, an important feature in analysis and applications.

Equivalent Metrics

Two metrics on the same set are called topologically equivalent when they generate the same open sets.

Equivalent metrics may assign different numerical distances while agreeing on convergence, continuity, open sets, and closed sets.

On \(\mathbb{R}^n\), the standard metrics \(d_1\), \(d_2\), and \(d_\infty\) are topologically equivalent.

Isometries

A function

\[ f:(X,d_X)\to(Y,d_Y) \]

is an isometry if it preserves distance exactly:

\[ d_Y(f(x),f(y))=d_X(x,y) \]

for all \(x,y\in X\).

Isometries preserve the metric geometry of a space.

Contractions

A function \(T:X\to X\) is a contraction if there exists a constant \(q\) with

\[ 0\le q<1 \]

such that

\[ d(Tx,Ty)\le qd(x,y) \]

for all \(x,y\in X\).

A contraction strictly shrinks distances by a uniform factor.

The Contraction Mapping Principle

If \((X,d)\) is a nonempty complete metric space and \(T:X\to X\) is a contraction, then \(T\) has a unique fixed point \(x^*\in X\) satisfying

\[ T(x^*)=x^*. \]

Moreover, starting from any \(x_0\in X\) and repeatedly applying \(T\),

\[ x_{n+1}=T(x_n), \]

produces a sequence converging to \(x^*\).

Why the Contraction Principle Is Important

The contraction mapping principle converts a geometric shrinking condition into an existence-and-uniqueness theorem.

It appears throughout analysis in iterative equations, numerical methods, differential equations, integral equations, and dynamical systems.

Its logic depends critically on completeness: the successive approximations form a Cauchy sequence, and completeness guarantees that the limit remains inside the space.

Common Error: Verifying Only Some Metric Axioms

A distance formula is not a metric merely because it is nonnegative and symmetric.

All four metric axioms must hold, especially the triangle inequality.

When testing a proposed metric, check the axioms systematically rather than relying on geometric intuition.

Common Error: Confusing Open Balls with Closed Balls

The open ball uses the strict inequality

\[ d(x,y)<r, \]

while the closed ball uses

\[ d(x,y)\le r. \]

The boundary points whose distance equals \(r\) belong to the closed ball but not to the open ball.

Common Error: Assuming Closed and Bounded Always Means Compact

In \(\mathbb{R}^n\), closed and bounded sets are compact.

In a general metric space, however,

\[ \text{closed and bounded} \centernot\Longrightarrow \text{compact}. \]

The Heine–Borel theorem is a special finite-dimensional result, not a universal metric-space rule.

Common Error: Assuming Every Cauchy Sequence Converges

Every convergent sequence is Cauchy, but the reverse implication requires completeness.

In an incomplete space such as \(\mathbb{Q}\), a Cauchy sequence may aim at a limit that lies outside the space.

Common Error: Treating the Metric as Secondary

The metric is part of the definition of the space.

The notation

\[ (X,d) \]

records both the set and its distance rule. Changing the metric can change numerical distances and may change convergence, continuity, compactness, and other structural properties.

A Reliable Procedure for Checking a Metric

  1. State the set \(X\) and the proposed function \(d:X\times X\to[0,\infty)\).
  2. Verify nonnegativity.
  3. Show that \(d(x,y)=0\) occurs exactly when \(x=y\).
  4. Verify symmetry.
  5. Prove the triangle inequality.
  6. Only after all four checks conclude that \(d\) is a metric.

A Reliable Procedure for Proving a Set Is Open

  1. Choose an arbitrary point \(x\) in the set.
  2. Measure or estimate the distance from \(x\) to the relevant boundary.
  3. Choose a radius \(r>0\) small enough to avoid leaving the set.
  4. Show that \(d(x,y)<r\) implies \(y\) remains in the set.
  5. Conclude that every point has an open ball contained in the set.

A Reliable Procedure for Sequence Convergence

  1. Identify the proposed limit \(x\).
  2. Write the distance \(d(x_n,x)\).
  3. Estimate that distance in terms of \(n\).
  4. Given \(\varepsilon>0\), choose \(N\) so the estimate is below \(\varepsilon\) whenever \(n\ge N\).
  5. Conclude that \(x_n\to x\).

A Reliable Procedure for Proving Completeness

  1. Begin with an arbitrary Cauchy sequence \((x_n)\) in the space.
  2. Use the structure of the space to identify a candidate limit.
  3. Prove the sequence converges to that candidate.
  4. Verify that the candidate belongs to the space.
  5. Conclude that every Cauchy sequence converges within the space.

Example: A Metric Verification

On \(\mathbb{R}\), define

\[ d(x,y)=2|x-y|. \]

Nonnegativity, identity of indiscernibles, and symmetry follow immediately from absolute value.

For the triangle inequality,

\[ d(x,z)=2|x-z| \le2(|x-y|+|y-z|) \]

and therefore

\[ d(x,z)\le d(x,y)+d(y,z). \]

Hence \(d\) is a metric.

Example: A Formula That Is Not a Metric

Consider

\[ d(x,y)=(x-y)^2 \]

on \(\mathbb{R}\).

It is nonnegative, symmetric, and zero exactly when \(x=y\), but the triangle inequality can fail.

For \(x=0\), \(y=1\), and \(z=2\),

\[ d(0,2)=4, \]

while

\[ d(0,1)+d(1,2)=1+1=2. \]

Since \(4\nleq2\), this formula is not a metric.

Example: Convergence in the Discrete Metric

Suppose \(X\) has the discrete metric and

\[ x_n\to x. \]

Choose \(\varepsilon=1/2\). Eventually,

\[ d(x_n,x)<\frac12. \]

But in the discrete metric the only distance smaller than \(1/2\) is zero. Therefore, eventually

\[ x_n=x. \]

A sequence converges in the discrete metric exactly when it is eventually constant.

Example: A Complete Function Space

Let

\[ C([a,b]) \]

denote the continuous real-valued functions on \([a,b]\). With the metric

\[ d_\infty(f,g)= \max_{x\in[a,b]}|f(x)-g(x)|, \]

this space is complete.

A Cauchy sequence in this metric is uniformly Cauchy, hence converges uniformly to a function, and the uniform limit of continuous functions is continuous.

Practice 1: Metric Axioms

On \(\mathbb{R}\), determine whether

\[ d(x,y)=|x-y|+1 \]

is a metric.

Practice 2: Euclidean Distance

Find the Euclidean distance between

\[ x=(1,-2,3) \]

and

\[ y=(4,2,3). \]

Practice 3: Taxicab Distance

Find \(d_1(x,y)\) for the same two points

\[ x=(1,-2,3), \qquad y=(4,2,3). \]

Practice 4: Open Ball on the Real Line

Describe the open ball

\[ B_3(5) \]

in \(\mathbb{R}\) with the usual metric.

Practice 5: Open or Closed?

In \(\mathbb{R}\) with the usual metric, classify

\[ [1,4] \]

as open, closed, both, or neither.

Practice 6: Closure

In \(\mathbb{R}\), find the closure of

\[ A=(0,1). \]

Practice 7: Metric Convergence

In \(\mathbb{R}^2\) with the Euclidean metric, determine the limit of

\[ x_n=\left(\frac1n,2+\frac1n\right). \]

Practice 8: Discrete-Metric Convergence

Let \(X=\mathbb{Z}\) with the discrete metric and define

\[ x_n=n. \]

Does \((x_n)\) converge in \(X\)?

Practice 9: Completeness

Explain why the closed interval

\[ [0,1] \]

with the usual metric is complete.

Practice 10: Contraction

On \(\mathbb{R}\), consider

\[ T(x)=\frac{x+4}{3}. \]

Show that \(T\) is a contraction and find its fixed point.

Practice Answers

Practice 1

The formula is not a metric because

\[ d(x,x)=|x-x|+1=1\ne0. \]

It fails the identity-of-indiscernibles requirement.

Practice 2

\[ d_2(x,y) =\sqrt{(1-4)^2+(-2-2)^2+(3-3)^2} \] \[ =\sqrt{9+16} =\boxed{5}. \]

Practice 3

\[ d_1(x,y) =|1-4|+|-2-2|+|3-3| \] \[ =3+4+0 =\boxed{7}. \]

Practice 4

By definition,

\[ B_3(5)=\{x\in\mathbb{R}:|x-5|<3\}. \]

Therefore,

\[ \boxed{B_3(5)=(2,8)}. \]

Practice 5

The interval \([1,4]\) contains its two limit endpoints, and its complement

\[ (-\infty,1)\cup(4,\infty) \]

is open. Therefore,

\[ \boxed{[1,4]\text{ is closed but not open}.} \]

Practice 6

The endpoints \(0\) and \(1\) are limit points of \((0,1)\). Thus

\[ \boxed{\overline{(0,1)}=[0,1]}. \]

Practice 7

Each coordinate converges:

\[ \frac1n\to0, \qquad 2+\frac1n\to2. \]

Hence

\[ \boxed{x_n\to(0,2)}. \]

Practice 8

A convergent sequence in the discrete metric must eventually equal its limit. The sequence

\[ 1,2,3,4,\ldots \]

is never eventually constant. Therefore it does not converge.

Practice 9

The real line \(\mathbb{R}\) is complete, and \([0,1]\) is a closed subset of \(\mathbb{R}\). Every closed subset of a complete metric space is complete. Hence

\[ \boxed{[0,1]\text{ is complete}.} \]

Practice 10

For \(x,y\in\mathbb{R}\),

\[ |T(x)-T(y)| =\left|\frac{x-y}{3}\right| =\frac13|x-y|. \]

Thus \(T\) is a contraction with constant \(q=1/3\). A fixed point satisfies

\[ x=\frac{x+4}{3}. \]

Therefore,

\[ 3x=x+4, \qquad 2x=4, \qquad \boxed{x=2}. \]

What to Remember

A metric abstracts the essential behavior of distance.

\[ \boxed{(X,d)} \]

becomes a metric space once \(d\) satisfies nonnegativity, identity of indiscernibles, symmetry, and the triangle inequality.

Where We Go Next

Metric spaces give analysis a general language of distance, but many spaces used in applied mathematics possess additional algebraic structure. In particular, spaces of vectors allow us to measure not only distances between points but also the size of individual vectors and the angles between them.

The next lesson, Normed and Inner-Product Spaces, adds vector-space structure to the metric viewpoint and develops norms, induced metrics, inner products, orthogonality, Cauchy–Schwarz, and the foundations of Banach and Hilbert spaces.