Introduction to Topology

Topology is a branch of mathematics that studies properties of spaces that are preserved under continuous deformations such as stretching, bending, and twisting, but not tearing or gluing. It's often described as "rubber sheet geometry" because it focuses on properties that remain unchanged when shapes are stretched or deformed.

Unlike geometry, which studies rigid shapes with fixed measurements, topology concerns itself with more abstract properties like connectedness, compactness, and continuity.

Key Concepts in Topology

1. Topological Spaces

A topological space is a set X together with a collection τ of subsets of X (called open sets) that satisfy the following axioms:

Both the empty set ∅ and X itself belong to τ
Any union of members of τ belongs to τ
Any finite intersection of members of τ belongs to τ

Example 1: Discrete Topology

On any set X, the collection of all subsets of X forms a topology called the discrete topology. In this topology, every subset is open.

For instance, if X = {a, b, c}, then τ = {∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}.

Example 2: Indiscrete Topology

The collection τ = {∅, X} forms the indiscrete or trivial topology. This is the minimal possible topology on X.

2. Continuity

A function f: X → Y between topological spaces is continuous if the preimage of every open set in Y is open in X.

Example: Continuous Function

Consider f: ℝ → ℝ given by f(x) = x². To show this is continuous in the standard topology, we need to verify that for any open interval (a,b) in ℝ, the preimage f⁻¹((a,b)) is open in ℝ.

For a > 0, f⁻¹((a,b)) = (-√b, -√a) ∪ (√a, √b), which is a union of open intervals, hence open.

For a ≤ 0, f⁻¹((a,b)) = (-√b, √b), which is an open interval, hence open.

3. Connectedness

A topological space X is connected if it cannot be represented as the union of two disjoint non-empty open sets.

Example: Connected vs. Disconnected

The real line ℝ is connected, whereas the set {0, 1} with the discrete topology is disconnected, as it can be written as the union of {0} and {1}, both of which are open in the discrete topology.

4. Compactness

A topological space X is compact if every open cover of X has a finite subcover.

Example: Compact Space

The closed interval [0,1] in ℝ is compact. By the Heine-Borel theorem, a subset of ℝⁿ is compact if and only if it is closed and bounded.

The open interval (0,1) is not compact, as the open cover {(1/n, 1-1/n) : n ≥ 2} has no finite subcover.

5. Homeomorphism

A homeomorphism is a continuous bijection with a continuous inverse. Two spaces are topologically equivalent (homeomorphic) if there exists a homeomorphism between them.

Example: Homeomorphic Spaces

A circle and a square are homeomorphic. Intuitively, you can continuously deform one into the other without tearing or gluing.

A coffee mug and a donut (torus) are homeomorphic, as they both have exactly one hole. This is often cited as a classic example in topology.

Types of Topological Spaces

1. Metric Spaces

A metric space is a set X with a distance function d: X × X → ℝ that satisfies:

d(x,y) ≥ 0 for all x,y ∈ X, with equality if and only if x = y
d(x,y) = d(y,x) for all x,y ∈ X (symmetry)
d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X (triangle inequality)

Example: Euclidean Space

ℝⁿ with the Euclidean distance d(x,y) = √(Σ(xᵢ-yᵢ)²) is a metric space.

2. Hausdorff Spaces

A topological space X is Hausdorff if for any distinct points x,y ∈ X, there exist disjoint open sets U and V such that x ∈ U and y ∈ V.

Example: Hausdorff Space

Any metric space is Hausdorff. For example, in ℝ with the standard topology, for any two distinct points a and b, we can take the open intervals (a-ε, a+ε) and (b-ε, b+ε) where ε = |b-a|/3, which are disjoint.

3. Manifolds

An n-dimensional manifold is a topological space where every point has a neighborhood homeomorphic to ℝⁿ.

Example: Manifolds

A circle is a 1-dimensional manifold, as locally it looks like a line segment.

A sphere is a 2-dimensional manifold, as locally it looks like a plane.

The torus (donut shape) is also a 2-dimensional manifold.

Problem-Solving Techniques in Topology

1. Proving Continuity

To prove a function f: X → Y is continuous, you can:

Show that the preimage of every open set in Y is open in X
Show that the preimage of every basis element of Y is open in X
Use the sequential characterization: a function is continuous if and only if it preserves convergent sequences

Example: Proving Continuity

Let's prove that f: ℝ → ℝ defined by f(x) = 3x + 2 is continuous.

For any open interval (a,b) in ℝ, we have:

f⁻¹((a,b)) = {x ∈ ℝ : a < 3x + 2 < b}

= {x ∈ ℝ : (a-2)/3 < x < (b-2)/3}

= ((a-2)/3, (b-2)/3)

This is an open interval in ℝ, therefore open. Thus, f is continuous.

2. Analyzing Connectedness

To determine if a space is connected:

Try to express it as a union of two disjoint non-empty open sets
Use the fact that the continuous image of a connected space is connected
Use the intermediate value theorem for connected subsets of ℝ

Example: Proving Connectedness

Let's prove that the interval [0,1] is connected.

Suppose [0,1] = A ∪ B, where A and B are disjoint non-empty open sets in the subspace topology.

Let a ∈ A and b ∈ B. Without loss of generality, assume a < b.

Let c = sup{x ∈ A : x ≤ b}. Then c must be in either A or B.

If c ∈ A, then since A is open in [0,1], there exists ε > 0 such that (c-ε, c+ε) ∩ [0,1] ⊆ A. But this means there are points in A greater than c, contradicting the definition of c.

If c ∈ B, similarly, we get a contradiction.

Therefore, [0,1] cannot be written as a union of two disjoint non-empty open sets, so it's connected.

3. Checking Compactness

To prove a space is compact:

Show that every open cover has a finite subcover
For subsets of ℝⁿ, use the Heine-Borel theorem (a subset is compact if and only if it's closed and bounded)
Use the fact that the continuous image of a compact space is compact

Example: Proving Compactness

To prove that [0,1] is compact, we can use the Heine-Borel theorem since [0,1] is closed and bounded in ℝ.

Alternatively, let {Uα}α∈I be an open cover of [0,1]. Define:

S = {x ∈ [0,1] : [0,x] can be covered by finitely many Uα}

Clearly, 0 ∈ S (as 0 is in some Uα), so S is non-empty.

S is bounded above by 1, so it has a supremum s.

We can show that s ∈ S and s = 1, which proves that [0,1] can be covered by finitely many Uα.

4. Finding Homeomorphisms

To prove two spaces X and Y are homeomorphic:

Construct a bijection f: X → Y
Prove that f is continuous
Prove that f⁻¹ is continuous

Example: Finding a Homeomorphism

Let's prove that the open interval (0,1) is homeomorphic to ℝ.

Define f: (0,1) → ℝ by f(x) = tan(π(x-1/2))

This function is bijective: as x approaches 0, f(x) approaches -∞, and as x approaches 1, f(x) approaches ∞.

f is continuous as it's a composition of continuous functions (tan and a linear function).

The inverse f⁻¹: ℝ → (0,1) given by f⁻¹(y) = (1/π)arctan(y) + 1/2 is also continuous.

Therefore, (0,1) and ℝ are homeomorphic.

Famous Topological Objects and Theorems

1. Möbius Strip

A Möbius strip is a surface with only one side and one edge. It can be created by taking a strip of paper, giving it a half-twist, and connecting the ends.

2. Klein Bottle

A Klein bottle is a non-orientable surface that has no inside or outside. It can be thought of as a "bottle" whose neck passes through its side without creating a hole and connects to the bottom.