Topology asks what survives when distance and angle are no longer the main language.

A topological space is one of the simplest structures in higher mathematics and one of the most powerful. Instead of beginning with lengths, coordinates or equations, topology begins by deciding which subsets count as open. From that single choice we can define neighbourhoods, continuity, closure, boundary, convergence patterns, connectedness, compactness and the idea that two spaces have the same shape in the topological sense.

This guide is the first BTT Mathematics Atlas article for R22 · Topology. It is written as a bridge from sets, functions and proof into university-level topology. The goal is not to memorise vocabulary. The goal is to learn how a small list of axioms generates a large mathematical world, how to test examples, and how to avoid importing geometric assumptions that the topology never gave you.

Return routes: BTT Mathematics Hub · Functions and graphs · Calculus · Geometry and measurement.

The simple answer

A topology on a set X is a collection of subsets of X that we agree to call open. The collection must satisfy three rules: the empty set and the whole set X are open; any union of open sets is open; and any finite intersection of open sets is open.

That definition may look almost too small. Its strength comes from what follows. Once the open sets are known, a function can be defined to be continuous without mentioning a formula, derivative or distance: a function f : X → Y is continuous when the inverse image of every open set in Y is open in X.

The familiar idea from calculus—nearby inputs should lead to nearby outputs—becomes a structural statement about how open sets pull back through a function. This definition works for ordinary real-valued functions, geometric spaces, function spaces and many spaces that have no natural ruler at all.

1. What must be known first

Sets, subsets, unions, intersections and complements.
Functions, domains, codomains, images and inverse images.
Basic proof methods: direct proof, contradiction and counterexample.
Some familiarity with intervals on the real line is helpful.

The most important prerequisite is the inverse image. If f : X → Y and V is a subset of Y, then f⁻¹(V) means all points x in X whose outputs f(x) lie in V. This notation does not require f to have an inverse function. Topological continuity is built from inverse images because inverse images preserve unions and intersections exactly.

2. The topology axioms

Let X be a set. A topology τ on X is a family of subsets of X satisfying:

∅ ∈ τ and X ∈ τ.
If {U_i} is any family of members of τ, then ⋃U_i ∈ τ.
If U₁, …, U_n ∈ τ, then U₁ ∩ … ∩ U_n ∈ τ.

The pair (X, τ) is called a topological space. Members of τ are the open sets.

Notice the asymmetry. Arbitrary unions are required, but only finite intersections. An infinite intersection of open sets can fail to be open. In the ordinary real line, every interval (−1/n, 1/n) is open, but their intersection over positive integers n is {0}, which is not open in the usual topology.

Worked example 1: a topology on three points

Let X = {a, b, c} and consider τ = {∅, {a}, {a, b}, X}. Is τ a topology?

The empty set and X are included. Unions of the listed sets remain on the list: for example {a} ∪ {a, b} = {a, b}. Finite intersections also remain on the list: {a} ∩ {a, b} = {a}. Therefore τ is a topology.

But note what this topology says. There is an open set containing a but not c, and an open set containing b but not c. There is no open set containing c except X itself. The three points are not topologically interchangeable.

Worked example 2: a tempting non-example

Let X = {1, 2, 3} and σ = {∅, {1, 2}, {2, 3}, X}. This is not a topology because {1, 2} ∩ {2, 3} = {2}, and {2} is missing from σ. One failed axiom is enough.

3. The coarsest and finest topologies

Every set X has at least two obvious topologies.

The indiscrete topology is {∅, X}. It has as few open sets as possible.
The discrete topology is the power set P(X). Every subset is open.

A topology τ₁ is coarser than τ₂ when τ₁ ⊆ τ₂. Equivalently, τ₂ is finer than τ₁. Finer means more open sets, not necessarily “better”. Changing the topology changes which functions are continuous and which sets have topological properties.

There is a useful direction test. If you make the domain topology finer, continuity becomes easier because more inverse images are allowed to be open. If you make the codomain topology finer, continuity becomes harder because there are more open sets whose inverse images must be checked.

4. The usual topology on the real line

For R, the ordinary topology can be described by open intervals. A set U ⊆ R is open when every point x in U has some radius ε > 0 such that (x − ε, x + ε) is contained in U.

This definition comes from the usual distance |x − y|, but once the open sets are identified, topology can proceed without repeatedly mentioning distance. Open intervals form a basis for the usual topology: every open set is a union of open intervals.

5. Bases: generating many open sets from a smaller family

A basis B for a topology on X is a collection of subsets satisfying two local coverage rules. First, every point of X lies in at least one basis element. Second, whenever x lies in B₁ ∩ B₂ for basis elements B₁ and B₂, there is a basis element B₃ containing x and contained in B₁ ∩ B₂.

The topology generated by B consists of all unions of basis elements. This gives a practical way to define a topology without listing every open set.

Worked example 3: half-open intervals

On R, take basis elements of the form [a, b) with a < b. They generate a topology called the lower limit topology or Sorgenfrey line. It is finer than the usual topology because every ordinary open interval can be written as a union of half-open basis intervals. Yet the spaces behave differently in important ways. The underlying set is still R; the topology changed.

6. Neighbourhoods

A set N is a neighbourhood of x if N contains an open set U with x ∈ U ⊆ N. Some books use “open neighbourhood” when they specifically require N itself to be open. The distinction matters, so check the local convention.

Neighbourhood language lets us speak locally. A property holds near x if it holds on some neighbourhood of x. In metric spaces this resembles “inside some small ball”, but topology does not assume a radius exists.

7. Closed sets, closure, interior and boundary

A set F ⊆ X is closed if X \ F is open. A set can be both open and closed; such a set is called clopen. The empty set and X are always clopen.

The closure cl(A), often written Ā, is the smallest closed set containing A. Equivalently, x belongs to cl(A) if every neighbourhood of x meets A.

The interior int(A) is the largest open set contained in A. A point x is an interior point of A if some neighbourhood of x lies entirely inside A.

The boundary ∂A is cl(A) \ int(A). A point is on the boundary when every neighbourhood of the point meets both A and its complement.

Worked example 4: the interval [0, 1)

In R with the usual topology, let A = [0, 1). Then int(A) = (0, 1), cl(A) = [0, 1], and ∂A = {0, 1}. The point 0 belongs to A but is not interior; the point 1 does not belong to A but is still a boundary point and belongs to the closure.

This is a useful correction to a common misconception: the boundary of a set is not merely “the edge points that are in the set”. Boundary is determined by neighbourhoods, not membership alone.

8. Continuity by inverse images

Let (X, τ_X) and (Y, τ_Y) be topological spaces. A function f : X → Y is continuous if f⁻¹(V) is open in X for every open set V in Y.

This is the central definition. It generalises the ε–δ definition from real analysis. In metric spaces the two definitions agree.

Worked example 5: x² is continuous topologically

Take f : R → R given by f(x) = x² with the usual topology. To prove continuity using topology, we could rely on the known ε–δ theorem that polynomials are continuous. But the topological content is this: whenever V is open in the output line, f⁻¹(V) must be open in the input line.

For example, if V = (1, 4), then f⁻¹(V) = (−2, −1) ∪ (1, 2), which is open. One example does not prove continuity, but it shows how inverse images behave. The theorem for polynomials guarantees the same for every open V.

Worked example 6: every map from a discrete space is continuous

If X has the discrete topology and f : X → Y is any function, then for every open V in Y, the set f⁻¹(V) is some subset of X. Every subset of X is open, so f is continuous. This is why a finer domain topology tends to make continuity easier.

Worked example 7: maps into an indiscrete space

If Y has the indiscrete topology {∅, Y}, then every map f : X → Y is continuous. There are only two open sets to test, and their inverse images are ∅ and X, which are always open.

9. Equivalent tests for continuity

The open-set definition has several equivalent forms. For f : X → Y, the following are equivalent:

The inverse image of every open set in Y is open in X.
The inverse image of every closed set in Y is closed in X.
For every subset A of X, f(cl(A)) ⊆ cl(f(A)).
For every point x and every neighbourhood V of f(x), there is a neighbourhood U of x such that f(U) ⊆ V.

These formulations are not different kinds of continuity. They are different handles on the same property. In proofs, choose the handle that matches the available information.

10. Why inverse images are used, not images

A continuous function does not have to send open sets to open sets. For instance, f : R → R given by f(x) = x² sends the open interval (−1, 1) to [0, 1), which is not open in R.

Continuity uses inverse images because inverse images cooperate perfectly with set operations:

f⁻¹(⋃V_i) = ⋃f⁻¹(V_i)
f⁻¹(V ∩ W) = f⁻¹(V) ∩ f⁻¹(W)
f⁻¹(Y \ V) = X \ f⁻¹(V)

That structural compatibility makes the topology axioms pull back cleanly.

11. Composition and restrictions

If f : X → Y and g : Y → Z are continuous, then g ∘ f : X → Z is continuous. The proof is almost automatic: for an open set W in Z, (g ∘ f)⁻¹(W) = f⁻¹(g⁻¹(W)). The inner inverse image is open because g is continuous; the outer inverse image is open because f is continuous.

If A is a subspace of X and f : X → Y is continuous, then the restriction f|_A : A → Y is continuous when A carries the subspace topology. These simple closure properties let larger constructions be assembled from smaller continuous pieces.

12. The subspace topology

If A ⊆ X and X has topology τ, the subspace topology on A consists of all sets A ∩ U where U is open in X.

A set can be open in the subspace A without being open in X. For example, [0, 1) is not open in R, but it is open in the subspace [0, 2] because [0, 1) = [0, 2] ∩ (−1, 1).

Always ask: open in which space? Topological adjectives are relative to the ambient topology.

13. Homeomorphism: when two spaces are topologically the same

A homeomorphism f : X → Y is a bijection such that f is continuous and f⁻¹ is continuous. If a homeomorphism exists, X and Y are called homeomorphic.

Homeomorphic spaces have the same topological structure. They may look different geometrically. An open interval (0, 1) is homeomorphic to R. One explicit map is x ↦ tan(π(x − 1/2)). It stretches the interval without tearing or gluing it and sends it bijectively onto the whole real line.

A continuous bijection need not be a homeomorphism. The inverse must also be continuous. Later articles will show conditions—such as compact domain and Hausdorff codomain—under which a continuous bijection automatically becomes a homeomorphism.

14. Topological properties and invariants

A topological property is a property preserved by homeomorphism. Examples include compactness, connectedness, the number of connected components and many separation properties. Length, angle, area and differentiability are not purely topological because they need additional structure.

This distinction is the reason topology is sometimes described informally as studying shape under continuous deformation. The phrase is useful as intuition but incomplete as a definition. Topology is not literally about rubber sheets; it is about spaces and continuous maps.

15. Common misconceptions

“Open means it has no boundary.” False. An open set may have a nonempty boundary; (0, 1) has boundary {0, 1}.
“Closed means bounded.” False. R is closed in itself and unbounded.
“A set cannot be both open and closed.” False. ∅ and X always are; disconnected spaces may have nontrivial clopen sets.
“Continuity means the image of every open set is open.” False. Continuity is defined by inverse images.
“A topology gives distances.” Not necessarily. Some topologies come from metrics; others do not.
“Open is absolute.” False. A set is open relative to a specified topology.
“A continuous bijection is automatically a homeomorphism.” False without additional hypotheses.

16. A proof pattern worth learning

Many early topology proofs follow a simple operating pattern:

State the object and the topology explicitly.
Choose the definition that matches the target claim.
Take an arbitrary point or arbitrary open set.
Use the topology axioms or the defining structure.
Return to the exact definition and close the argument.

For a continuity proof, the template is often: “Let V be open in Y. We show f⁻¹(V) is open in X.” The rest of the proof depends on the particular spaces and map.

17. Worked proof: the identity map between two topologies

Suppose τ₁ and τ₂ are topologies on the same set X, with τ₁ ⊆ τ₂. Consider the identity map id : (X, τ₂) → (X, τ₁).

Take an open set U in the codomain topology τ₁. Because τ₁ ⊆ τ₂, U is also open in the domain topology τ₂. Since id⁻¹(U) = U, the inverse image is open. Therefore id is continuous.

In the other direction, id : (X, τ₁) → (X, τ₂) need not be continuous. This example turns the “finer-domain continuity is easier” heuristic into a proof.

18. Worked classification on a two-point set

Let X = {a, b}. How many topologies are there? Every topology must contain ∅ and X. The remaining possible subsets are {a} and {b}. We may include neither, include only {a}, include only {b}, or include both. Each choice satisfies the axioms. So there are four topologies:

{∅, X}
{∅, {a}, X}
{∅, {b}, X}
{∅, {a}, {b}, X}

This tiny example is useful because it shows topology is additional structure placed on a set. The point set alone does not determine the open sets.

19. Practice set

On X = {1,2,3}, decide whether {∅,{1},X} is a topology.
On the same X, decide whether {∅,{1},{2},X} is a topology.
Give the closure, interior and boundary of (0,1] in the usual topology on R.
Explain why arbitrary intersections of open sets are not required to be open.
Prove that arbitrary unions of closed sets need not be closed by giving an example in R.
Let X be discrete. Prove every subset is both open and closed.
Let Y be indiscrete. Prove every map X → Y is continuous.
Show that a constant function X → Y is continuous for arbitrary topological spaces.
Prove the composition of two continuous functions is continuous.
Let A ⊆ X. Explain why A is open in the subspace topology on A.
In R, find an open set U such that U ∩ [0,1] = [0,1/2).
Show that cl(A) is closed.
Show that A ⊆ cl(A).
Give an example of a clopen subset that is neither ∅ nor the whole space.
Explain why a homeomorphism must preserve openness.
Is f(x)=x² : R→[0,∞) a homeomorphism? Explain.
Is f(x)=x³ : R→R a homeomorphism? Explain.
Describe the topology generated on R by all intervals (a,∞).

20. Answers and checks

Yes. The required unions and finite intersections stay in the family.
No. {1} ∪ {2} = {1,2} is missing.
Interior (0,1), closure [0,1], boundary {0,1}.
The topology axioms only guarantee finite intersections. In R, ⋂_n≥1(−1/n,1/n)={0}, not open.
For example F_n={1/n} is closed, but ⋃F_n={1,1/2,1/3,…} is not closed because 0 is a missing limit point.
In a discrete space every subset is open; its complement is also open, so the subset is closed.
The only open sets in Y are ∅ and Y, whose inverse images are ∅ and X.
If f(x)=y₀, the inverse image of an open V is X when y₀∈V and ∅ otherwise.
Use (g∘f)⁻¹(W)=f⁻¹(g⁻¹(W)).
A=A∩X and X is open in X.
One choice is U=(−1,1/2).
It is the intersection of all closed sets containing A, and arbitrary intersections of closed sets are closed.
Every closed set used in the intersection contains A.
In a discrete space with at least two points, any proper nonempty subset is clopen.
If U is open in X and h:X→Y is a homeomorphism, then U=(h⁻¹)⁻¹(U), so h(U) is open in Y.
No. It is not injective.
Yes. It is a continuous bijection with continuous inverse y↦∛y.
The generated topology contains ∅, R and rays (a,∞), together with arbitrary unions of such rays; these unions are again rays of the same form or R.

21. What this article has established

You should now be able to test whether a family of subsets is a topology, distinguish finer and coarser topologies, work with bases and neighbourhoods, compute closure/interior/boundary in standard examples, use the inverse-image definition of continuity, recognise subspace topology, and explain what a homeomorphism means.

The next layer is to reintroduce a controlled notion of distance and ask what additional structure it provides. Continue with Metric Spaces and Examples, then move to compactness, connectedness, products and quotients.

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