Teaching objectives
Topology lab for second-year undergraduates. A topology is a list of sets called open sets, and nothing more: the lab has the student build it by hand before using it. First on a set of three points (●, ■, ★) drawn on a triangle, where each open set is a potato that surrounds its points; then on the line and the plane, with the topologies that appear in every course: standard, Sorgenfrey, right rays, cofinite, discrete and lexicographic.
The examples and challenges differ from those in the course notes: the lab serves as extra practice and does not repeat the theory.
What you learn
- To decide whether a family is a topology and, if it is not, to point out which union or which intersection is missing.
- To build the generated topology of some given sets: the smallest one containing them, closing under unions and intersections.
- That on three points there are 29 topologies and only 9 types once the points are renamed.
- Closed sets as complements of open sets, and that open and closed are not opposites.
- What a base is (the two conditions), which topology it generates and which is the smallest base of a finite space.
- To recognise whether a family of subsets of ℝ is a topology and to find the counterexample when it is not.
- The five reference topologies of the line by the shape of their basic open sets, and to decide whether a set is open, closed, both or neither in each.
- The subspace topology: a set is not simply open, it is open in a universe.
- Comparing topologies: finer = more open sets; the criterion of fitting a basic set of one inside a basic set of the other (“the brick”).
- The lexicographic order of the plane, its intervals, and that its topology is strictly finer than the usual one.
Key mathematical ideas
- Open means being on the list. There is nothing geometric in the definition: the geometry appears when the list is chosen.
- What is missing is not a choice. If a family contains A and B, the topology must contain A ∪ B and A ∩ B; that is why the generated topology is unique.
- Comparing topologies is comparing neighbourhoods. 𝒯 ⊂ 𝒯′ if and only if around each point, inside each basic set of 𝒯, there fits a basic set of 𝒯′ containing it. Discs, squares and diamonds give the same topology because they fit inside one another.
- In Sorgenfrey the right endpoints are the troublemakers. (a; b) is a union of intervals [x; b), but no (c; d) containing a fits inside [a; b).
- The map of the line. trivial ⊂ {SRD, cofinite} ⊂ standard ⊂ Sorgenfrey ⊂ discrete, with SRD and cofinite incomparable.
- In the lexicographic plane the neighbourhoods are vertical segments. The vertical line x = 0 is open and the horizontal one is not.
Sources and credits
The notation and order follow the U-tad Topology course notes (topics 3 and 4). General references: J. R. Munkres, Topology (2nd ed., 2000), chapters 2 and 3; L. A. Steen and J. A. Seebach, Counterexamples in Topology (1978), for the topologies of the line and the ordered square.
Classroom notes
- “Collect the three-point topologies” works well as a competition in pairs: who finds the 9 types first?
- “The brick criterion” is understood at once when projected, with the animation that covers the disc with squares.
- “Covering the ball with segments” shows in a single picture why the lexicographic topology is finer than the usual one: it is worth asking beforehand whether the line x = 0 is open.
Room-by-room contents
Room 1 · Open means being on the list
Why open sets were invented without distances (Fréchet 1906, Hausdorff 1914; robots and topological data analysis). Definition of a topology. X = {●, ■, ★} on a triangle, with the open sets drawn as potatoes. Five families: decide whether each is a topology and, if not, point out a set that has to be there.
Student tasks
- Decide whether five families are topologies.
- Point out a required set that is missing.
Room 2 · Completing to a topology
Some sets that must be open are given and the student switches on the ones that are needed. Check marks in amber the unions and intersections that are missing and in white where they come from. The answer is the generated topology, the smallest one.
Student tasks
- Complete five families to the smallest topology (4 to 8 open sets).
Room 3 · Collect the three-point topologies
Gallery of the 9 types of topology on three points (29 in total). Each new topology you build goes into its slot; repeated ones, up to renaming, light up.
Student tasks
- Recognise two topologies of the same type.
- Collect all 9 types.
- See the full gallery: 9 types, 29 topologies once renamings are counted.
Room 4 · Closed sets: the complements
F is closed if X ∖ F is open. Closed sets appear as purple potatoes. Open and closed are not opposites: some sets are open and closed at once and others are neither open nor closed.
Student tasks
- Mark all the closed sets of a topology.
- Mark those that are both open and closed.
- Mark those that are neither.
- Build a topology in which {●} is both open and closed.
Room 5 · Bases
The two conditions for a base and the generated topology (all unions). Minimal base of a finite space: the smallest open set of each point. Bridge to ℝ: {[a;b] : a < b} is not a base.
Student tasks
- Decide whether a family is a base.
- Complete a family to a base with as few sets as possible.
- Build the topology generated by a base.
- Find the minimal base.
- Choose the family in ℝ that is not a base.
Room 6 · Topologies on the line?
Six families of subsets of ℝ (symmetric intervals, closed rays, particular point, intervals, cocountable, left rays). If it is not a topology, say what fails and the line animates the counterexample.
Student tasks
- Decide six families.
- Say what fails in the ones that are not topologies.
Room 7 · Building open sets from a base
The probe shows the basic open set of the standard topology, Sorgenfrey, SRD (right rays), cofinite and discrete. Then one topology per challenge: with its base, can each set be built as a union of basic sets? A set is open exactly when it can be built; the ones that cannot fail at one specific endpoint that no basic set manages to surround.
Student tasks
- Try the five topologies with the probe.
- Decide, in each of the five topologies, which three sets can be built as a union of basic sets and which cannot.
Room 8 · Open or closed?
Open = union of basic sets; closed = its complement is open: two separate checks. Each challenge is one set in three topologies, with one label per row (open, closed, open and closed, neither open nor closed). A button switches the line between the set and its complement.
Student tasks
- Classify four sets, each in three topologies.
Room 9 · Open in a universe
Subspace topology: A is open in Y if A = U ∩ Y for some open set U of ℝ. Y is the light strip on the axis itself; the student drags the endpoints of the interval U and sees U ∩ Y live next to the set to be built (A, or Y ∖ A for closed), with = or ≠.
Student tasks
- Eight cases: build the set as U ∩ Y (solved automatically when they match) or press “It is not open / closed in Y” when it cannot be done.
Room 10 · Finer, coarser
𝒯′ is finer than 𝒯 if it has all of its open sets. Pairs of three-point topologies and the minimal open set of a point, which in the finer one is equal or smaller.
Student tasks
- Compare four pairs.
- Find the minimal open set of ■ in two topologies.
Room 11 · The brick criterion
Comparison criterion: inside each basic set of 𝒯 there fits a basic set of 𝒯′ containing the point. Fit squares, discs and diamonds inside one another near the edge; animation that covers the disc with squares and the square with discs.
Student tasks
- Fit four bricks.
- Say what it proves that they fit in both directions.
Room 12 · Sorgenfrey is finer
(1;3) is a union of intervals [x;3). The other way round, no (c;d) containing 1 fits inside [1;3).
Student tasks
- Fit two bricks [x;c).
- Try to fit a (c;d) inside [1;3).
- Decide the relation and where it fails.
Room 13 · Below the standard topology
The rays (a;+∞) and the complements of finite sets are open in the standard topology. SRD and cofinite are incomparable.
Student tasks
- Fit a brick inside a ray.
- Count the intervals of ℝ minus three points.
- Give an open set of each that is not open in the other.
- Classify four sets.
Room 14 · Dictionary order
The lexicographic order on the plane: the first coordinate rules. What an interval looks like: a piece of a column, whole columns and a piece of another column.
Student tasks
- Sort two lists of points.
- Decide which candidates lie in an interval.
- Recognise an interval inside a vertical line.
Room 15 · Covering the ball with segments
Vertical segments are enough as a base. Animation of the disc covered by vertical segments; no disc fits inside a segment. The line x = 0 is open and the line y = 0 is not.
Student tasks
- Fit two segments inside a disc.
- Decide whether a disc fits inside a segment.
- Decide whether four sets are open in the lexicographic plane.
Room 16 · The map of topologies
The fineness map of trivial, cofinite, SRD, standard, Sorgenfrey and discrete.
Student tasks
- Compare eight pairs.
- Place the six in the diagram.
- Count those that are finer than SRD.
Rooms to project
The most striking ones to show and discuss in class.