Topology is a subject where examples do not merely illustrate the theory; they are the theory.
Most definitions were created to capture a class of examples and to exclude another class with a precise boundary.
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So if you want to work fluently in topology, you need a dependable method for building spaces and maps on demand.
This piece lays out a practical recipe for constructing examples and counterexamples without relying on a memorized zoo.
Start by choosing which feature you want to control
Most example-building begins by deciding which of these properties you want to enforce or break:
- Compactness
- Connectedness and path connectedness
- Hausdorff separation and related separation axioms
- Countability properties: first countable, second countable, separable
- Local properties: locally compact, locally connected, locally path connected
- Metrizability
Once you choose the target feature, you pick a construction that is known to interact strongly with that feature.
The four construction engines that generate most examples
Subspaces: enforce constraints without changing the ambient world
Subspaces are the most conservative way to build examples.
- If you want to preserve Hausdorffness, taking a subspace keeps it.
- If you want to preserve compactness, taking a closed subspace keeps it.
So a standard move is:
- Start in a familiar Hausdorff space like $\mathbb{R}^n$ or a product of intervals.
- Cut out a \subset with the property you want.
Classic uses:
- Totally disconnected compact sets inside $[0,1]$.
- Spaces that are connected but not path connected as carefully chosen subspaces of $\mathbb{R}^2$.
- Locally complicated sets where local structure breaks naive intuition.
Subspaces are also how you build “tame but nontrivial” examples that remain metrizable.
Products: amplify dimension, create new compactness behavior
Products are the simplest way to create higher-dimensional phenomena from one-dimensional pieces.
- Products preserve compactness when each factor is compact.
- Products preserve Hausdorffness when each factor is Hausdorff.
- Products often change countability properties dramatically.
The product topology is designed to be the coarsest topology making all projections continuous, so it behaves well with mapping arguments.
The box topology is a useful contrast tool:
- It often breaks compactness and countability properties.
- It is a standard way to build counterexamples where naive “coordinatewise” reasoning fails.
If you want a space where “local finiteness” matters, products are usually the right engine.
Quotients: create identifications and force global glue
Quotients are where topology becomes visibly geometric.
You can build circles, spheres, projective spaces, wedges, cones, suspensions, and cell complexes by identifying points.
A quotient starts with a space $X$ and an equivalence relation $\sim$.
The quotient map $q:X\to X/\sim$ is continuous by definition, but separation properties can change drastically.
Quotients are the main tool when you want \to:
- Create non-Hausdorff spaces from Hausdorff ones.
- Build compact spaces by identifying boundaries of compact sets.
- Force new loops and higher-dimensional holes.
They are also the place where you must learn to diagnose when the quotient is well-behaved, for example when equivalence classes are closed and the relation is compatible with compactness.
Topology-by-design: specify opens indirectly
Sometimes you do not want to start from a metric or an embedding. You want \to design a topology with a specific behavior.
Two standard methods are:
- Basis generation: declare a collection of sets to be basic opens and check the basis axioms.
- Initial and final topologies: declare which maps should be continuous and take the coarsest or finest topology that enforces that.
This is how you get:
- The discrete and indiscrete topologies.
- The cofinite and cocountable topologies.
- The lower limit topology on $\mathbb{R}$, which is a classic source of subtle countability behavior.
- The Sierpiński space, which is a minimal test object for continuity and for order-topological ideas.
These designed topologies are often the quickest way to build sharp counterexamples.
A practical workflow for building counterexamples
A counterexample usually needs two ingredients:
- A space with a property you expect.
- A carefully chosen map or topology that breaks the property you are trying to prove.
Instead of searching randomly, proceed with a controlled workflow:
- Decide which theorem you want to violate and identify the hypothesis you will remove.
- Choose a construction that is sensitive to that hypothesis.
- Build the simplest possible space where the sensitivity is visible.
- Verify the desired properties directly from the definition that actually applies in that context.
For example, \to violate “continuous bijection implies homeomorphism,” remove compactness and use a continuous bijection where inverse continuity would require control of closed sets.
To violate “quotient of Hausdorff is Hausdorff,” build a quotient that forces two distinct points to share neighborhoods after identification.
How to build compactness and non-compactness on purpose
Compactness can be produced reliably by:
- Closed subspaces of compact spaces.
- Finite products of compact spaces.
- Quotients of compact spaces.
Non-compactness can be produced reliably by:
- Removing a limit point from a compact space.
- Infinite coproduct-like constructions that create infinitely many disjoint opens.
- Switching from product topology \to a finer topology like the box topology on an infinite product.
A useful mental picture is:
- Compactness is robust under “glue and restrict” operations.
- Non-compactness is often created by allowing escape or by increasing the topology so that more open covers exist.
A table: which constructions preserve which properties
| Construction | Preserves compactness | Preserves Hausdorffness | Often breaks countability | Typical use |
|—|—|—|—|—|
| Subspace | Yes for closed subspaces | Yes | Sometimes | Controlled, metrizable examples |
| Product topology | Yes for compact factors | Yes for Hausdorff factors | Can | Build higher-dimensional spaces, test uniformity |
| Box topology | Frequently no | Yes for Hausdorff factors | Yes | Build counterexamples in infinite products |
| Quotient | Yes if starting space compact | Not always | Can | Build geometric objects, glue points, create loops |
| Designed topology | By design | By design | By design | Sharp minimal counterexamples |
This table is not a substitute for proofs, but it tells you where to look when you want a property to survive.
A toolbox of example families worth mastering
You do not need hundreds of examples, but you do need a few families you can modify.
Order topologies
Any totally ordered set has an order topology.
These spaces are excellent for:
- Building non-metrizable but still understandable spaces.
- Creating spaces with unusual local bases.
- Testing compactness via order-completeness behavior.
The long line is a famous example of how order and local Euclidean-looking structure can coexist with non-second-countability.
One-point compactifications
If $X$ is locally compact and Hausdorff and non-compact, you can often form the one-point compactification $X^*$ by adding a point at infinity with neighborhoods that are complements of compact sets.
This construction is a controlled way \to:
- Turn non-compact into compact while preserving Hausdorffness.
- Encode “escape to infinity” as convergence \to a single point.
It is also a clean way to build compact spaces with desired behavior at infinity.
Wedges, cones, suspensions
These identification constructions let you create spaces with predictable connectedness and loop behavior.
- The wedge sum is a minimal way to attach two spaces at a point.
- Cones contract spaces in a controlled way.
- Suspensions shift loop-like features into higher-dimensional features.
Even if you do not compute invariants explicitly, these constructions let you reason about which kinds of paths and neighborhoods exist.
CW complexes and cell attachments
CW complexes provide an example-building framework that is simultaneously flexible and structured.
- You build a space by attaching cells in increasing dimension.
- Many invariants become computable by inductive arguments.
When you want examples that are complicated globally but tame locally, CW complexes are often the right language.
Verifying properties: pick the correct definition for the context
A common mistake is verifying properties using a characterization that does not apply.
Examples:
- Using sequences to argue about compactness in a non-metrizable space.
- Using “closed and bounded” outside $\mathbb{R}^n$.
- Using path-based intuition in spaces that are connected but not path connected.
A reliable habit is:
- First identify whether your space is metric, first countable, or second countable.
- Then choose the strongest characterization that is valid.
For metric spaces, sequence arguments are fine. For general spaces, return to open sets, covers, and the finite intersection property.
Building examples is building intuition
When you can build spaces systematically, topology stops feeling like a list of axioms and starts feeling like controlled engineering.
- If you want a phenomenon, choose a construction engine that amplifies it.
- If you want to avoid a pathology, choose a construction engine that preserves the property you care about.
That is the real skill behind reading and writing topology proofs. The best proofs are usually the ones where the author silently chose the example-building method that makes the conclusion inevitable.
One concrete build: the torus as a quotient you can visualize
A reliable way to generate a nontrivial compact space is to start with a compact rectangle $[0,1]\times [0,1]$ and identify boundary points.
- Identify $(0,y)\sim (1,y)$ for all $y\in[0,1]$.
- Identify $(x,0)\sim (x,1)$ for all $x\in[0,1]$.
The quotient space is the torus.
This example is a template because it lets you check properties directly:
- Compactness survives because the starting square is compact and quotients of compact spaces are compact.
- Hausdorffness survives here because the equivalence relation is closed and the identifications are “tame” in a precise sense.
- Connectedness survives because continuous images of connected spaces are connected.
Once you know this template, you can build variants that break Hausdorffness by changing the equivalence relation in a way that makes equivalence classes accumulate.
A controlled non-Hausdorff quotient as a warning example
Start with two copies of $\mathbb{R}$, call them $\mathbb{R}_a$ and $\mathbb{R}_b$.
Identify every nonzero point $x\neq 0$ in $\mathbb{R}_a$ with the corresponding point in $\mathbb{R}_b$, but keep the two origins $0_a$ and $0_b$ distinct.
The resulting space is the “line with two origins.”
It is locally like $\mathbb{R}$ away from the origin, and it is still connected, but it is not Hausdorff: any neighborhoods of $0_a$ and $0_b$ intersect because they both contain points close to zero that have been identified.
This is a compact illustration of what quotients can do:
- Quotients are excellent for building geometry.
- Quotients can also quietly destroy separation, so verifying Hausdorffness is not optional.
A final habit: build maps at the same time as spaces
In topology, an example is usually a pair $(X,f)$ rather than a bare space.
Once you build a space, immediately ask what natural maps it comes with:
- Inclusion maps from subspaces.
- Projection maps from products.
- Quotient maps from identifications.
- Collapse maps that contract a subspace \to a point.
Many theorems are really statements about how these canonical maps behave.
If you train yourself to build the map alongside the space, your examples will automatically align with the proof techniques you will later need.
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