Homeomorphism | Malone Math | Malone Global University

The Central Idea

A homeomorphism is a function

\[ f:X\to Y \]

satisfying four conditions:

\(f\) is a function from \(X\) to \(Y\);
\(f\) is bijective;
\(f\) is continuous;
\(f^{-1}\) is continuous.

If such a function exists, then

\[ \boxed{X\cong Y}, \]

meaning that \(X\) and \(Y\) are homeomorphic.

Topological Equivalence

Homeomorphic spaces are considered equivalent from the viewpoint of topology.

They may differ in:

size;
angle;
curvature;
position;
coordinate representation.

But they share the same underlying topological structure.

What a Homeomorphism Preserves

A homeomorphism preserves every topological property.

Examples include:

connectedness;
path connectedness;
compactness;
number of connected components;
openness and closedness under the induced correspondence;
boundary structure in the intrinsic topologies;
genus for appropriate surfaces;
Euler characteristic for suitable spaces.

What a Homeomorphism Need Not Preserve

A homeomorphism does not need to preserve ordinary geometric quantities.

It may change:

distance;
angle;
area;
volume;
curvature;
straightness in an ambient Euclidean picture.

These are geometric rather than purely topological properties.

Why Bijectivity Matters

A homeomorphism must pair every point of \(X\) with exactly one point of \(Y\), and every point of \(Y\) must be reached.

Bijectivity combines:

injectivity: distinct points of \(X\) remain distinct;
surjectivity: every point of \(Y\) is represented.

Without bijectivity, the function may collapse points together or omit part of the target space.

Why Continuity Matters

Continuity prevents the map from making abrupt topological jumps.

One topological definition says that

\[ f:X\to Y \]

is continuous when the preimage of every open set in \(Y\) is open in \(X\).

Thus, if \(V\subseteq Y\) is open, then

\[ \boxed{f^{-1}(V)\text{ is open in }X}. \]

Why the Inverse Must Also Be Continuous

A continuous bijection need not automatically be a homeomorphism.

The inverse map must also preserve the topology continuously.

This prevents the forward map from hiding a topological distortion that becomes discontinuous when reversed.

Equivalent Open-Set Description

A bijection

\[ f:X\to Y \]

is a homeomorphism if and only if it sends open sets of \(X\) to open sets of \(Y\) and is continuous.

In that case, \(f\) is both a continuous map and an open map.

Equivalently, it is a continuous bijection whose inverse is continuous.

Homeomorphisms Also Preserve Closed Sets

Since homeomorphisms preserve open sets and complements,

if \(F\subseteq X\) is closed, then

\[ f(F) \]

is closed in \(Y\).

Thus homeomorphisms preserve the full open-closed structure of the spaces.

A Simple Example: Translation

Define

\[ f:\mathbb{R}\to\mathbb{R} \]

\[ f(x)=x+5. \]

This map is bijective and continuous.

Its inverse is

\[ f^{-1}(y)=y-5, \]

which is also continuous.

Therefore \(f\) is a homeomorphism.

A Scaling Example

Define

\[ f:\mathbb{R}\to\mathbb{R} \]

\[ f(x)=3x. \]

Its inverse is

\[ f^{-1}(y)=\frac{y}{3}. \]

Both maps are continuous, so

\[ \boxed{\mathbb{R}\cong\mathbb{R}} \]

under this scaling.

Intervals of Different Lengths

The closed intervals

\[ [0,1] \]

and

\[ [5,20] \]

are homeomorphic.

One homeomorphism is

\[ f(x)=5+15x. \]

Its inverse is

\[ f^{-1}(y)=\frac{y-5}{15}. \]

Exact length is irrelevant topologically.

General Linear Homeomorphism Between Closed Intervals

\[ a and
\[ c then
\[ f:[a,b]\to[c,d] \]
defined by
\[ \boxed{ f(x) = c+\frac{d-c}{b-a}(x-a) } \]
is a homeomorphism.

All Open Intervals Are Homeomorphic

If
\[ a and
\[ c then
\[ (a,b)\cong(c,d). \]
A linear transformation provides the homeomorphism.

Therefore exact endpoint positions and interval lengths are not topological invariants.

An Open Interval Is Homeomorphic to the Real Line

A more surprising example is
\[ (0,1)\cong\mathbb{R}. \]
One possible homeomorphism is
\[ f(x)=\tan\left(\pi x-\frac{\pi}{2}\right). \]
As \(x\) approaches \(0\), \(f(x)\) approaches \(-\infty\), and as \(x\) approaches \(1\), \(f(x)\) approaches \(+\infty\).

A Rational Formula for \((0,1)\) and \(\mathbb{R}\)

Another useful homeomorphism is
\[ f(x)=\frac{2x-1}{x(1-x)} \]
after suitable monotonic verification, although trigonometric formulas are often cleaner.

The important lesson is that boundedness is not a topological invariant.

Boundedness Is Not Topological

The interval
\[ (0,1) \]
is bounded in \(\mathbb{R}\).

The real line
\[ \mathbb{R} \]
is unbounded.

Yet they are homeomorphic.

Therefore boundedness is not preserved by homeomorphism.

Closed Interval vs. Open Interval

The spaces
\[ [0,1] \]
and
\[ (0,1) \]
are not homeomorphic.

One reason is compactness.

The closed interval is compact, while the open interval is not.

Since compactness is a topological invariant,
\[ \boxed{[0,1]\not\cong(0,1)}. \]

The Circle and an Interval Are Not Homeomorphic

Consider
\[ S^1 \]
and
\[ [0,1]. \]
Both are compact, connected, and path connected.

Those invariants do not distinguish them.

But removing an interior point from \([0,1]\) disconnects the interval, while removing any one point from \(S^1\) leaves a connected space.

Using Cut Points to Distinguish Spaces

A point \(p\) of a connected space is a cut point if removing it disconnects the space.

Interior points of
\[ [0,1] \]
are cut points.

No point of the circle
\[ S^1 \]
is a cut point.

Therefore the circle and interval cannot be homeomorphic.

The Sphere and the Cube Surface

The surface of a cube is homeomorphic to the sphere.

The cube has corners and flat faces, while the sphere is smooth.

Those geometric differences do not matter topologically.

Both are connected closed orientable surfaces of genus \(0\).

The Torus and a Coffee Mug

A mug with one handle is a familiar informal model of a surface homeomorphic to a torus.

Both possess one handle-like topological feature.

Their genus is
\[ g=1. \]
The analogy illustrates that topology ignores geometric shape while preserving structural features such as handles.

Sphere vs. Torus

The sphere and torus are not homeomorphic.

One reason is genus:
\[ g(S^2)=0, \qquad g(T^2)=1. \]
Another is Euler characteristic:
\[ \chi(S^2)=2, \qquad \chi(T^2)=0. \]
Different invariant values prove non-homeomorphism.

Homeomorphism and Connectedness

If
\[ X\cong Y, \]
then
\[ X\text{ connected} \iff Y\text{ connected}. \]
A homeomorphism cannot turn one connected component into several separated components.

Homeomorphism and Path Connectedness

Homeomorphisms preserve paths.

If
\[ \gamma:[0,1]\to X \]
is a path and
\[ f:X\to Y \]
is a homeomorphism, then
\[ f\circ\gamma \]
is a path in \(Y\).

Therefore path connectedness is preserved.

Homeomorphism and Compactness

If \(X\) is compact and
\[ f:X\to Y \]
is a homeomorphism, then \(Y=f(X)\) is compact because continuous images of compact spaces are compact.

Applying the same reasoning to \(f^{-1}\) gives the converse.

Thus compactness is preserved in both directions.

Homeomorphism and Components

A homeomorphism maps connected components of \(X\) onto connected components of \(Y\).

Therefore homeomorphic spaces have the same number and pattern of connected components.

This gives a quick method for disproving many proposed homeomorphisms.

Example: One Interval vs. Two Intervals

Compare
\[ X=(0,1) \]
with
\[ Y=(0,1)\cup(2,3). \]
The first space is connected.

The second has two connected components.

Therefore,
\[ \boxed{X\not\cong Y}. \]

Homeomorphism Is an Equivalence Relation

Homeomorphism satisfies the three properties of an equivalence relation.

reflexive;

symmetric;

transitive.

Therefore topological spaces can be grouped into equivalence classes under homeomorphism.

Reflexivity

Every space is homeomorphic to itself.

The identity map
\[ \operatorname{id}_X:X\to X \]
defined by
\[ \operatorname{id}_X(x)=x \]
is bijective and continuous, and its inverse is itself.

Hence,
\[ \boxed{X\cong X}. \]

Symmetry

If
\[ f:X\to Y \]
is a homeomorphism, then
\[ f^{-1}:Y\to X \]
is also a homeomorphism.

Therefore,
\[ X\cong Y \Longrightarrow Y\cong X. \]

Transitivity

Suppose
\[ X\cong Y \]
through a homeomorphism \(f\), and
\[ Y\cong Z \]
through a homeomorphism \(g\).

Then
\[ g\circ f:X\to Z \]
is a homeomorphism.

Hence,
\[ X\cong Z. \]

Constructing a Homeomorphism

To prove that two spaces are homeomorphic, one usually constructs an explicit map.

A complete proof typically checks:

the function is well-defined;

the function is injective;

the function is surjective;

the function is continuous;

the inverse is continuous.

Sometimes the Inverse Can Be Written Explicitly

For example, let
\[ f:(0,\infty)\to\mathbb{R} \]
be
\[ f(x)=\ln x. \]
The inverse is
\[ f^{-1}(y)=e^y. \]
Both maps are continuous, so
\[ \boxed{(0,\infty)\cong\mathbb{R}}. \]

Continuous Bijective Maps from Compact to Hausdorff Spaces

There is an important shortcut.

Suppose \(X\) is compact, \(Y\) is Hausdorff, and
\[ f:X\to Y \]
is a continuous bijection.

Then \(f\) is automatically a homeomorphism.
\[ \boxed{ X\text{ compact},\; Y\text{ Hausdorff},\; f\text{ continuous bijection} \Longrightarrow f\text{ homeomorphism} }. \]

Why the Compact-to-Hausdorff Shortcut Works

Closed subsets of compact spaces are compact.

Continuous images of compact sets are compact.

Compact subsets of Hausdorff spaces are closed.

Therefore a continuous bijection
\[ f:X\to Y \]
from compact \(X\) to Hausdorff \(Y\) maps closed sets to closed sets.

Hence the inverse map is continuous.

A Continuous Bijection Need Not Be a Homeomorphism

Consider the identity function
\[ f:X\to Y \]
where \(X\) and \(Y\) have the same underlying set but different topologies.

The function can be continuous and bijective in one direction while its inverse is not continuous.

This shows why continuity of the inverse must be part of the definition.

Example Using Different Topologies

Let \(X\) be a set with the discrete topology and let \(Y\) be the same set with a coarser topology.

The identity map
\[ \operatorname{id}:X\to Y \]
is continuous because every open set of \(Y\) is open in the discrete topology on \(X\).

But the inverse may fail to be continuous because not every subset is open in \(Y\).

Thus a continuous bijection need not preserve the topology in both directions.

Embeddings

An embedding is a map that identifies one space homeomorphically with its image inside another space.

A map
\[ f:X\to Y \]
is an embedding when
\[ f:X\to f(X) \]
is a homeomorphism, where \(f(X)\) carries the subspace topology from \(Y\).

Homeomorphism vs. Embedding

A homeomorphism compares two entire spaces.

An embedding places one space inside another without changing its intrinsic topology.

For example, the unit circle
\[ S^1 \]
is embedded in
\[ \mathbb{R}^2. \]

Homeomorphism vs. Isometry

An isometry preserves distance exactly.

A homeomorphism need not.

Therefore,
\[ \boxed{ \text{isometry} \Longrightarrow \text{homeomorphism} } \]
in the usual metric-topological setting, but
\[ \boxed{ \text{homeomorphism} \centernot\Longrightarrow \text{isometry} }. \]

Homeomorphism vs. Diffeomorphism

A diffeomorphism is a smooth bijection with smooth inverse between smooth manifolds.

Smoothness is stronger than continuity.

Therefore every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.

Homeomorphism vs. Homotopy Equivalence

A homotopy equivalence is generally weaker than a homeomorphism.

Homeomorphic spaces have exactly the same topology.

Homotopy-equivalent spaces may have different local structure while sharing broader deformation behavior.

This distinction becomes important in algebraic topology.

Topological Invariants as Obstructions

To prove two spaces are homeomorphic, one usually needs an actual homeomorphism.

To prove they are not homeomorphic, it is often enough to find one topological invariant that differs.

This gives a powerful strategy:
\[ \boxed{ \text{different invariant} \Longrightarrow \text{not homeomorphic} }. \]

Using Compactness as an Obstruction

Compare
\[ [0,1] \]
and
\[ \mathbb{R}. \]
The first is compact.

The second is not.

Therefore,
\[ \boxed{[0,1]\not\cong\mathbb{R}}. \]

Using Connectedness as an Obstruction

Compare
\[ (0,1) \]
with
\[ (0,1)\cup(2,3). \]
The first is connected.

The second is disconnected.

Therefore they are not homeomorphic.

Using Number of Components as an Obstruction

If \(X\) has three connected components and \(Y\) has five, then
\[ \boxed{X\not\cong Y}. \]
A homeomorphism cannot create or destroy components.

Using Genus as an Obstruction

For closed orientable surfaces,
\[ g \]
is invariant under homeomorphism.

Therefore a torus with
\[ g=1 \]
cannot be homeomorphic to a double torus with
\[ g=2. \]

Using Euler Characteristic as an Obstruction

If
\[ \chi(X)\ne\chi(Y), \]
then \(X\) and \(Y\) cannot be homeomorphic.

For example,
\[ \chi(S^2)=2 \]
while
\[ \chi(T^2)=0. \]
Hence
\[ S^2\not\cong T^2. \]

Matching Invariants Do Not Automatically Prove Homeomorphism

Suppose two spaces are both:

compact;

connected;

path connected.

That does not necessarily mean they are homeomorphic.

Invariants are excellent for disproving equivalence, but matching a few invariants may not be enough to prove equivalence.

Example: Circle and Closed Interval

Both
\[ S^1 \]
and
\[ [0,1] \]
are compact, connected, and path connected.

Yet they are not homeomorphic because their cut-point behavior differs.

This illustrates why multiple invariants may be needed.

Local Structure Matters

Homeomorphism preserves not only global properties but local topological behavior.

A point with a neighborhood resembling an open interval cannot correspond under a homeomorphism to a point whose neighborhoods have fundamentally different local structure.

This idea becomes central in the study of manifolds.

Interior Points and Endpoint-Type Points

In the interval
\[ [0,1], \]
interior points locally resemble open intervals.

The endpoints have one-sided neighborhood structure.

A homeomorphism of \([0,1]\) onto itself must send endpoints to endpoints, because endpoint-type local structure is preserved.

Boundary Is Relative to the Ambient Space

Care is required when discussing boundary as an invariant.

The boundary of a subset depends on the ambient space.

A homeomorphism between spaces preserves their intrinsic topological structure, but it does not automatically preserve the boundary of an embedding in some unrelated larger ambient space.

Always specify which topology is being compared.

Common Error: Thinking Homeomorphic Means Congruent

Congruent figures have the same geometric size and shape.

Homeomorphic spaces need only have the same topological structure.

A circle of radius \(1\) and a circle of radius \(100\) are homeomorphic even though they are not congruent.

Common Error: Thinking Homeomorphic Means Similar

Similarity preserves shape up to scale.

Homeomorphism allows much more general deformation.

A square boundary and a circle are homeomorphic even though one cannot be obtained from the other by a single similarity transformation.

Common Error: Forgetting Bijectivity

A continuous map that collapses several points into one is not a homeomorphism.

For example,
\[ f:[-1,1]\to[0,1] \]
defined by
\[ f(x)=x^2 \]
is continuous and onto, but not injective because
\[ f(x)=f(-x). \]
Therefore it is not a homeomorphism.

Common Error: Forgetting Surjectivity

A continuous injective map may fail to cover the entire target.

If part of \(Y\) is omitted, then the function cannot be a homeomorphism from \(X\) onto \(Y\).

It may instead define an embedding into \(Y\).

Common Error: Checking Only the Forward Map

A bijective continuous map is not automatically a homeomorphism.

Unless a theorem such as the compact-to-Hausdorff criterion applies, one must also verify continuity of
\[ f^{-1}. \]

Common Error: Using a Geometric Difference That Is Not Topological

Different lengths, angles, curvature, or areas do not prove that spaces are non-homeomorphic.

To disprove homeomorphism, use a genuine topological invariant or a topological structural argument.

Common Error: Assuming Same Euler Characteristic Is Enough

Equal Euler characteristic does not always imply homeomorphism.

The torus and Klein bottle both satisfy
\[ \chi=0, \]
but they differ in orientability.

A matching invariant is not necessarily a complete classification.

Common Error: Ignoring the Topologies

Homeomorphism is a relationship between topological spaces, not merely sets of points.

The same underlying set equipped with different topologies may fail to be homeomorphic through the identity map.

The topology is part of the object.

A Reliable Procedure for Proving Homeomorphism

Define a candidate map \(f:X\to Y\).

Verify that \(f\) is well-defined.

Prove injectivity.

Prove surjectivity.

Prove continuity.

Find or characterize \(f^{-1}\).

Prove that \(f^{-1}\) is continuous.

Conclude \(X\cong Y\).

A Reliable Procedure for Proving Non-Homeomorphism

List topological properties of each space.

Compare connectedness and number of components.

Compare compactness.

Compare path connectedness and cut-point behavior.

For surfaces, compare genus, orientability, and Euler characteristic.

Use local structure when global invariants agree.

One differing invariant is enough to conclude non-homeomorphism.

Example: Prove Two Intervals Are Homeomorphic

Show that
\[ [0,1]\cong[2,8]. \]
Define
\[ f(x)=2+6x. \]
Then \(f\) is continuous and bijective.

Its inverse is
\[ f^{-1}(y)=\frac{y-2}{6}, \]
which is continuous.

Therefore,
\[ \boxed{[0,1]\cong[2,8]}. \]

Example: Prove \((0,\infty)\cong\mathbb{R}\)

Define
\[ f(x)=\ln x. \]
This is continuous, injective, and surjective from
\[ (0,\infty) \]
onto
\[ \mathbb{R}. \]
Its inverse
\[ f^{-1}(y)=e^y \]
is continuous.

Hence,
\[ \boxed{(0,\infty)\cong\mathbb{R}}. \]

Example: Show \([0,1]\not\cong\mathbb{R}\)

The interval
\[ [0,1] \]
is compact.

The real line
\[ \mathbb{R} \]
is not compact.

Since compactness is preserved by homeomorphism,
\[ \boxed{[0,1]\not\cong\mathbb{R}}. \]

Example: Show \(\mathbb{R}\not\cong\mathbb{R}^2\)

Remove one point from each space.

The punctured real line
\[ \mathbb{R}\setminus\{0\} \]
is disconnected.

The punctured plane
\[ \mathbb{R}^2\setminus\{(0,0)\} \]
remains path connected.

Homeomorphisms preserve this deletion behavior under corresponding points, so
\[ \boxed{\mathbb{R}\not\cong\mathbb{R}^2}. \]

Practice 1: Definition

Complete the definition:

A homeomorphism is a __________, __________ function whose inverse is also __________.

Practice 2: Linear Homeomorphism

Find a homeomorphism
\[ f:[0,1]\to[10,20]. \]

Practice 3: Inverse

If
\[ f(x)=4x-3, \]
find \(f^{-1}(y)\).

Practice 4: Compactness Test

Can
\[ [0,1] \]
be homeomorphic to
\[ (0,1)? \]
Explain using compactness.

Practice 5: Connectedness Test

Can
\[ (0,1) \]
be homeomorphic to
\[ (0,1)\cup(2,3)? \]

Practice 6: Circle and Interval

Why are
\[ S^1 \]
and
\[ [0,1] \]
not homeomorphic even though both are compact and connected?

Practice 7: Sphere and Torus

Give one topological invariant that proves
\[ S^2\not\cong T^2. \]

Practice 8: Continuous Bijection

Is every continuous bijection automatically a homeomorphism?

Practice 9: Compact-to-Hausdorff Shortcut

Suppose \(X\) is compact, \(Y\) is Hausdorff, and
\[ f:X\to Y \]
is a continuous bijection.

What can be concluded?

Practice 10: Equivalence Relation

Name the three properties showing that homeomorphism is an equivalence relation.

Practice Answers

Practice 1

A homeomorphism is a bijective, continuous function whose inverse is also continuous.

Practice 2

One suitable map is
\[ \boxed{f(x)=10+10x}. \]
It maps \(0\) to \(10\) and \(1\) to \(20\), is continuous and bijective, and has a continuous inverse.

Practice 3

Starting with
\[ y=4x-3, \]
solve for \(x\):
\[ x=\frac{y+3}{4}. \]
Therefore,
\[ \boxed{f^{-1}(y)=\frac{y+3}{4}}. \]
Practice 4

No.

The closed interval \([0,1]\) is compact, while \((0,1)\) is not compact. Since compactness is preserved by homeomorphism,
\[ \boxed{[0,1]\not\cong(0,1)}. \]
Practice 5

No.

The interval \((0,1)\) is connected, while \((0,1)\cup(2,3)\) is disconnected.
\[ \boxed{(0,1)\not\cong(0,1)\cup(2,3)}. \]
Practice 6

Removing an interior point from \([0,1]\) disconnects it, while removing any one point from \(S^1\) leaves a connected space. Their cut-point behavior differs.

Practice 7

Genus distinguishes them:
\[ g(S^2)=0, \qquad g(T^2)=1. \]
Euler characteristic also works:
\[ \chi(S^2)=2, \qquad \chi(T^2)=0. \]
Practice 8

No. The inverse must also be continuous unless an additional theorem guarantees it.

Practice 9

The map is automatically a homeomorphism:
\[ \boxed{f:X\to Y\text{ is a homeomorphism}}. \]
Practice 10

Homeomorphism is:

reflexive;

symmetric;

transitive.

What to Remember

A homeomorphism is a bijective continuous map with continuous inverse.

Homeomorphic spaces are topologically equivalent.

Homeomorphisms preserve open and closed structure.

They preserve connectedness, path connectedness, compactness, and component structure.

For suitable surfaces, they preserve genus and Euler characteristic.

They do not need to preserve distance, angle, area, or curvature.

A continuous bijection need not be a homeomorphism unless its inverse is also continuous.

A continuous bijection from a compact space to a Hausdorff space is automatically a homeomorphism.

To prove homeomorphism, construct and verify a suitable map.

To disprove homeomorphism, find a topological invariant that differs.

The defining relationship is
\[ \boxed{ X\cong Y \iff \text{there exists a bijective continuous } f:X\to Y \text{ with continuous }f^{-1} }. \]
Homeomorphism is the formal language topology uses for same structure, different appearance.

Where We Go Next

Homeomorphism brings together many of the ideas developed so far. Open sets define continuity, connectedness and compactness provide invariants, and genus and Euler characteristic distinguish important surface types.

From here, topology can move toward deformation, quotient spaces, manifolds, homotopy, fundamental groups, and algebraic invariants. These tools expand the central classification question: when are two spaces topologically the same, and when can we prove that they are fundamentally different?