Continuity and Deformation

The Central Idea

A continuous map preserves the local topological organization of a space.

A deformation is a continuous family of maps that changes one configuration into another over time.

Continuity in Topology

Let

be a function between topological spaces.

The function is continuous if the preimage of every open set in \(Y\) is open in \(X\).

Why Preimages Appear in the Definition

Topology describes a space through its open sets, so a continuous function must respect open-set structure.

These identities make preimages naturally compatible with the topology axioms.

Continuity Through Closed Sets

A function \(f:X\to Y\) is continuous if and only if the preimage of every closed set in \(Y\) is closed in \(X\).

Continuity Through Neighborhoods

A map \(f:X\to Y\) is continuous at \(x\in X\) if for every neighborhood \(V\) of \(f(x)\), there exists a neighborhood \(U\) of \(x\) such that

Metric Continuity

In metric spaces, neighborhood continuity becomes the familiar \(\varepsilon\)-\(\delta\) definition.

Thus analytic continuity is a special case of topological continuity.

Continuity and Convergence

In metric spaces, if

and \(f\) is continuous, then

Continuous Images of Connected Spaces

If \(X\) is connected and \(f:X\to Y\) is continuous, then

A continuous map may collapse a connected space, but it cannot split its image into separated pieces.

Continuous Images of Path Connected Spaces

If \(X\) is path connected and \(f:X\to Y\) is continuous, then

A path \(\gamma\) in \(X\) becomes the path \(f\circ\gamma\) in the image.

Continuous Images of Compact Spaces

If \(X\) is compact and \(f:X\to Y\) is continuous, then

Composition of Continuous Maps

If \(f:X\to Y\) and \(g:Y\to Z\) are continuous, then

Homeomorphism as Reversible Continuity

A homeomorphism is a bijective continuous map whose inverse is continuous.

Homeomorphism represents exact equivalence at the topological level.

Deformation Is Weaker Than Homeomorphism

Topology also studies looser forms of equivalence.

Two spaces need not be homeomorphic in order to share the same broad deformation behavior.

The concept that formalizes continuous deformation between maps is homotopy.

Homotopy Between Maps

Let \(f,g:X\to Y\) be continuous maps.

A homotopy from \(f\) to \(g\) is a continuous function

such that

The Homotopy Parameter

The variable \(t\in[0,1]\) can be interpreted as time.

At each fixed \(t\),

is a map from \(X\) to \(Y\). The family \(H_t\) continuously changes \(f\) into \(g\).

Homotopic Maps

If a homotopy exists between \(f\) and \(g\), we write

This means the maps are homotopic, not necessarily equal.

A Straight-Line Homotopy

Suppose \(Y\subseteq\mathbb{R}^n\) is convex and \(f,g:X\to Y\) are continuous.

Then

gives a homotopy from \(f\) to \(g\).

Paths as Deformations of Points

A path

can be viewed as a continuous motion from \(\gamma(0)\) to \(\gamma(1)\).

This is the simplest form of deformation in topology.

Homotopy of Paths

Two paths with the same endpoints may be continuously deformed into one another while keeping the endpoints fixed.

Loops

A loop based at \(x_0\) is a path \(\gamma:[0,1]\to X\) satisfying

Homotopy classes of loops lead to the fundamental group.

Null-Homotopic Loops

A loop is null-homotopic if it can be continuously shrunk to the constant loop while remaining in the space.

Informally, the loop can collapse to a point without crossing a topological obstruction.

Simple Connectivity

A path connected space is simply connected if every loop is null-homotopic.

Equivalently, its fundamental group is trivial:

The Disk Is Simply Connected

Every loop in the disk \(D^2\) can be contracted to a point while remaining inside the disk.

The Circle Is Not Simply Connected

A loop traveling once around \(S^1\) cannot be contracted to a point while remaining on the circle.

Contractible Spaces

A space \(X\) is contractible if the identity map is homotopic to a constant map.

In other words, the entire space can be continuously shrunk to a point within itself.

Convex Sets Are Contractible

Let \(C\subseteq\mathbb{R}^n\) be convex and choose \(p\in C\).

At \(t=0\), \(H(x,0)=x\), and at \(t=1\), \(H(x,1)=p\).

Therefore every convex set is contractible.

Contractible Does Not Mean Homeomorphic to a Point

The disk \(D^2\) is contractible but is not homeomorphic to a one-point space.

Homotopy equivalence is therefore weaker than homeomorphism.

Homotopy Equivalence

Spaces \(X\) and \(Y\) are homotopy equivalent if there are continuous maps

such that

and

Notation for Homotopy Equivalence

We often write

when \(X\) and \(Y\) are homotopy equivalent.

Homeomorphism Implies Homotopy Equivalence

If

then

A homeomorphism preserves strictly more information.

The Converse Fails

Homotopy-equivalent spaces need not be homeomorphic.

but

Retractions

Let \(A\subseteq X\).

A retraction is a continuous map

satisfying

for every \(a\in A\).

Deformation Retracts

A subset \(A\subseteq X\) is a deformation retract of \(X\) if \(X\) can be continuously deformed onto \(A\) while \(A\) remains fixed.

A strong deformation retraction is a homotopy

such that

Why Deformation Retracts Matter

If \(A\) is a deformation retract of \(X\), then

The simpler subspace \(A\) has the same homotopy type as \(X\).

Annulus Deformation Retracts onto a Circle

Consider

Radial motion can move every point to a fixed intermediate circle while keeping that circle fixed.

The Punctured Plane Deformation Retracts onto a Circle

For

the radial map

projects every point onto \(S^1\).

A Disk Does Not Retract onto Its Boundary

There is no continuous retraction

that fixes every boundary point.

This fundamental result is closely connected with the Brouwer fixed-point theorem.

Isotopy

An isotopy is a deformation through embeddings or homeomorphisms, depending on context.

It is more restrictive than an arbitrary homotopy because each intermediate stage must preserve the relevant embedding structure.

Homotopy vs. Isotopy

A homotopy may allow intermediate maps to lose injectivity.

An isotopy keeps the relevant maps embedded or homeomorphic throughout.

Ambient Isotopy

In knot theory, two knots are equivalent when one can be carried to the other by an ambient isotopy of the surrounding space.

The knot remains embedded at every stage, so strands cannot pass through one another.

Why Homotopy Is Too Weak for Knot Equivalence

Ordinary homotopy can permit self-intersections during intermediate stages.

If strands are allowed to pass through one another, knotting information can disappear.

Knot theory therefore uses ambient isotopy instead.

Homotopy Invariants

A homotopy invariant is preserved under homotopy equivalence.

Examples include:

• fundamental group up to isomorphism;
• homology groups;
• cohomology groups;
• Euler characteristic for suitable finite-type spaces.

Euler Characteristic and Homotopy

For suitable finite cell complexes,

Compactness Is Not a Homotopy Invariant

The real line is contractible:

But \(\mathbb{R}\) is not compact while a point is compact.

Therefore compactness is not preserved by homotopy equivalence.

Dimension Is Not a Homotopy Invariant

A disk is homotopy equivalent to a point:

Their dimensions differ, so dimension is not a homotopy invariant.

Homotopy Forgets Local Structure

Homeomorphism preserves local neighborhood structure.

Homotopy equivalence can collapse large regions and ignore local dimension.

This makes homotopy especially useful for studying global holes and loop behavior.

Deformation and Holes

The disk is contractible because no essential loop obstruction remains.

The annulus is not contractible because a loop around its central hole cannot be contracted to a point while staying inside the annulus.

Circle vs. Annulus

The circle and annulus are not homeomorphic because their local dimensions and boundary structures differ.

But the annulus deformation retracts onto the circle.

Disk vs. Point

A disk and a point are not homeomorphic, but the disk contracts to any chosen point.

Sphere vs. Point

The sphere \(S^2\) is not contractible.

Its higher-dimensional topology prevents the entire sphere from shrinking continuously to a point within itself.

Homology provides one rigorous way to detect this obstruction.

Homotopy Classes

Topology often groups maps into homotopy classes.

The homotopy class of a map \(f\) is commonly written

Two maps belong to the same class if they are homotopic.

Homotopy Is an Equivalence Relation

Homotopy satisfies:

• reflexivity;
• symmetry;
• transitivity.

Therefore maps can be organized into equivalence classes under homotopy.

Reflexivity

Every map is homotopic to itself using the constant-in-time homotopy

Symmetry

If \(H\) is a homotopy from \(f\) to \(g\), then

runs the deformation backward.

Transitivity

If \(f\simeq g\) and \(g\simeq h\), the two homotopies can be reparameterized and concatenated to produce a homotopy from \(f\) to \(h\).

Continuity in Both Variables Matters

A homotopy

must be continuous jointly in \(x\) and \(t\).

It is not enough for separate frames to be continuous if the family jumps discontinuously as time changes.

Common Error: Thinking Continuity Means Differentiability

A continuous function need not be differentiable.

Topological continuity requires preservation of open-set structure, not derivatives.

Common Error: Thinking Continuous Means Injective

A continuous map may identify many points.

For example,

is continuous on \(\mathbb{R}\) but satisfies \(f(x)=f(-x)\).

Common Error: Thinking Homotopy Means Homeomorphism

A homotopy is a deformation between maps.

Intermediate maps need not be invertible.

Therefore homotopy is much weaker than homeomorphism.

Common Error: Thinking Homotopy Preserves Dimension

The disk is homotopy equivalent to a point.

Thus homotopy can ignore local dimension; homeomorphism cannot.

Common Error: Thinking Contractible Means Geometrically Tiny

Contractibility is a topological statement about a homotopy to a point.

It does not require preserving lengths, angles, areas, or rigid geometry.

Common Error: Thinking Every Retraction Is a Deformation Retraction

A retraction provides a continuous map \(r:X\to A\) fixing \(A\).

A deformation retraction additionally requires a continuous deformation from the identity to that retraction.

Common Error: Thinking Compactness Is a Homotopy Invariant

It is not.

but only the point is compact.

Common Error: Using Ordinary Homotopy for Knot Equivalence

Ordinary homotopy may permit intermediate self-intersections.

Knot equivalence requires ambient isotopy so the loop remains embedded throughout.

A Reliable Procedure for Testing Continuity

1. Identify the topologies on the domain and codomain.
2. Take an arbitrary open set in the codomain.
3. Compute or characterize its preimage.
4. Show that the preimage is open in the domain.
5. In metric settings, use an \(\varepsilon\)-\(\delta\) or sequential test when appropriate.

A Reliable Procedure for Building a Homotopy

1. Identify the starting map \(f\).
2. Identify the ending map \(g\).
3. Introduce \(t\in[0,1]\).
4. Construct \(H(x,t)\) with \(H(x,0)=f(x)\).
5. Verify \(H(x,1)=g(x)\).
6. Verify that \(H(x,t)\) remains inside the target space.
7. Prove that \(H\) is continuous.

A Reliable Procedure for Showing Contractibility

1. Choose a point \(p\in X\).
2. Construct a homotopy from \(\operatorname{id}_X\) to the constant map \(x\mapsto p\).
3. Verify the endpoint conditions.
4. Verify continuity.
5. Verify that every intermediate point remains inside \(X\).

Example: Contracting an Interval

Let \(X=[0,1]\).

Define

Then \(H(x,0)=x\) and \(H(x,1)=0\).

Example: Contracting a Disk

For

define

Since \(\|(1-t)x\|\le1\), the homotopy remains inside the disk.

Example: Why the Annulus Is Not Contractible

The annulus deformation retracts onto \(S^1\).

Since

the annulus has nontrivial loop structure and cannot be contractible.

Practice 1: Open-Set Continuity

Complete:

Practice 2: Composition

If \(f:X\to Y\) and \(g:Y\to Z\) are continuous, what can be concluded about \(g\circ f\)?

Practice 3: Homotopy Definition

State the domain and endpoint conditions for a homotopy from \(f:X\to Y\) to \(g:X\to Y\).

Practice 4: Contractible Interval

Give a homotopy contracting \([0,1]\) to \(0\).

Practice 5: Homeomorphism and Homotopy

Does \(X\cong Y\) imply \(X\simeq Y\)?

Practice 6: Converse

Does \(X\simeq Y\) imply \(X\cong Y\)?

Practice 7: Annulus

To what simpler space does an annulus deformation retract?

Practice 8: Punctured Plane

Practice 9: Homotopy Invariant

Is compactness a homotopy invariant?

Practice 10: Knot Deformation

Why is ordinary homotopy too weak to define equivalence of knots?

Practice Answers

Practice 1

Practice 2

Practice 3

A homotopy is a continuous map

with

Practice 4

Practice 5

Yes.

Practice 6

No. Homotopy equivalence is weaker than homeomorphism.

but the spaces are not homeomorphic.

Practice 7

Practice 8

Practice 9

No. A point is compact and \(\mathbb{R}\) is not, yet both are contractible.

Practice 10

Ordinary homotopy can allow self-intersections, effectively letting strands pass through one another. Knot equivalence requires ambient isotopy so the knot remains embedded throughout.

What to Remember

• A continuous map has open preimages of open sets.
• Continuity can also be expressed through neighborhoods, closed sets, or metric \(\varepsilon\)-\(\delta\) conditions.
• Continuous images preserve connectedness, path connectedness, and compactness.
• A homotopy is a continuous deformation between maps.
• Homotopy equivalence is weaker than homeomorphism.
• A contractible space is homotopy equivalent to a point.
• Convex subsets of Euclidean space are contractible.
• Deformation retracts provide simpler spaces with the same homotopy type.
• The annulus and punctured plane deformation retract onto \(S^1\).
• Isotopy is stronger than homotopy because embedding structure is preserved throughout.
• Knot equivalence uses ambient isotopy rather than ordinary homotopy.

but generally not conversely.

Where We Go Next

Continuity and deformation provide the bridge from elementary topology to algebraic topology. Homeomorphism asks whether spaces are exactly the same topologically, while homotopy asks whether their broad deformation behavior agrees.

From here, the natural next steps include fundamental groups, covering spaces, homotopy classes, deformation retracts, homology, cohomology, and algebraic topology. These tools convert deformation behavior into algebraic invariants that can distinguish increasingly subtle spaces.