2.1 Spaces and Continuity · Homology & Cohomology

…a set can be open, or closed, or both, or neither!

— James Munkres, Topology, 2nd ed. (2000)

The Prelude told the old joke that a topologist is someone who cannot tell a coffee cup from a doughnut. Behind the joke sits a real question. In what sense are a cup and a doughnut “the same”? And if they are, how could we ever prove that two shapes are different?

§0.1 The Shape of a Question gave a slogan: topology studies the properties of a shape that survive stretching, bending and squeezing, but not tearing or gluing. A slogan is not yet mathematics. To use it we need to say what a “shape” is and what it means to stretch one without tearing it. Two short definitions will do both — topological space and continuous map — and we will reach them by small steps, starting from something everyone already understands: distance.

Rubber-sheet geometry

Imagine drawing on a sheet of very stretchy rubber. Draw a closed loop, put a dot inside it and another dot outside it, and draw a curve that crosses the loop. Now grab the sheet and pull. Lengths change. Areas change. Straight lines go wavy, circles turn into blobs, right angles open out. Almost everything a geometry teacher would measure is destroyed.

And yet some things stubbornly refuse to change. The dot that was inside the loop is still inside it. The dot outside is still outside. The loop is still one closed curve, and the curve still crosses it the same number of times. No amount of stretching can change these facts — unless you tear the sheet, or press two parts of it together so hard that they stick. Try it in Figure 2.1.1.

A rubber sheet Drag the gold pins to stretch · then try Tear and Glue

Changes when you stretch

length of the loop 100%
area inside it 100%
shapes, angles, straightness

Survives any stretching

the rose point is inside the loop
the teal point is outside it
the violet curve crosses the loop 2 times

Figure 2.1.1 Stretching changes every measurement under “changes when you stretch”, but none of the facts under “survives any stretching”. Tearing the loop open destroys “inside”, and gluing two of its points together splits the inside in two. If you pull too hard the sheet folds over itself; that is not allowed either, because two different points would end up in the same place.

This is the starting point of topology. We want to keep track of exactly the information that survives stretching, and deliberately forget everything else. The facts that survive are all about nearness: which points are close to which, which regions touch, what is surrounded by what. Distances themselves are forgotten — the rubber can stretch them — but “being close” is not forgotten, because stretching never pulls neighbouring points apart into separate pieces.

So our plan is this. First we describe nearness with the help of distance, because distance is familiar. Then we notice that all the facts we care about can be phrased without mentioning any particular distance, using only certain sets called open sets. Finally we throw the distance away and keep only the open sets. What remains is a topological space: a set of points together with a precise notion of nearness, and nothing else.

Measuring distance: metric spaces

How far is it from $A$ to $B ?$ On a map, you would lay a ruler between them. In a city laid out on a grid, a taxi driver would give you a different answer, because the taxi cannot drive through buildings: it must go along streets, east–west and north–south. Both answers are perfectly reasonable ways to measure distance. Let us write them down.

A point of the plane is a pair of numbers $x = (x_{1}, x_{2}) .$ The straight-line distance between $x$ and $y = (y_{1}, y_{2})$ comes from Pythagoras’ theorem:

d_{2} (x, y) = (x_{1} - y_{1})^{2} + (x_{2} - y_{2})^{2} .

The taxicab distance adds up how far you must go east–west and how far north–south:

d_{1} (x, y) = ∣ x_{1} - y_{1} ∣ + ∣ x_{2} - y_{2} ∣ .

(Here $∣ t ∣,$ read “the absolute value of $t ”,$ means $t$ with its minus sign removed, if it has one: $∣ - 3 ∣ = 3$ and $∣ 3 ∣ = 3 .)$ There is even a third, the max distance $d_{\infty} (x, y) = max (∣ x_{1} - y_{1} ∣, ∣ x_{2} - y_{2} ∣),$ which is how many moves a king needs on a chessboard. The little subscripts $1, 2, \infty$ are just names.

What do these three have in common? Mathematicians answer such a question by listing the properties that every sensible distance should have, and then calling anything with those properties a distance. The list is short.

Each rule is common sense. Distances are never negative, and the only point at distance zero from you is yourself. The way there is as long as the way back. And the triangle inequality says that a detour through $y$ can never be a shortcut: going from $x$ to $z$ directly is at most as long as going via $y .$ All three of our distances obey all three rules. So does the distance along a line, $d (x, y) = ∣ x - y ∣$ for real numbers, and so does the straight-line distance in three-dimensional space or in any dimension $n,$ where $R^{n}$ (read “R n”) is the set of lists of $n$ real numbers.

A slightly mischievous example shows how loose the definition is. On any set, declare that the distance between two different points is always $1,$ and the distance from a point to itself is $0 .$ You can check that the three rules hold. This is the discrete metric: every point is at arm’s length from every other, and nothing is close to anything.

Three rulers for the plane

Figure 2.1.2 The taxi ride: from

A

B

it is

5

blocks as the crow flies but

7

by taxi, and every staircase route is equally long. The balls: the points at distance less than

1

from

x

form a diamond for the taxicab distance, a disk for the straight-line distance and a square for the max distance. Each fits inside the next.

The right half of Figure 2.1.2 introduces the most important shape in this chapter. The open ball of radius $r$ around a point $x$ is the set of all points less than $r$ away from $x :$

B (x, r) = {y \in X ∣ d (y, x) < r} .

(Read this as “the set of all $y$ in $X$ such that the distance from $y$ to $x$ is less than $r ”.)$ With the straight-line distance in the plane, $B (x, r)$ is a round disk without its rim — “open” because the rim, where the distance is exactly $r,$ is left out. With the taxicab distance the same formula gives a diamond, and with the max distance a square. On the real line, $B (x, r)$ is the interval from $x - r$ to $x + r,$ endpoints excluded. With the discrete metric, a ball of radius $\frac{1}{2}$ contains just its centre: everything else is at distance $1 .$

Open sets: room to wiggle

Here is the idea that unlocks everything. Think of a region of the plane — a pond, say — and a point inside it. Call the point comfortably inside if you can draw some disk around it, however small, that stays entirely in the pond. A frog sitting at such a point has room to wiggle in every direction without getting its feet wet.

Points well inside the pond are comfortably inside. What about a point exactly on the shore? If the shore belongs to the region, then every disk around a shore point, no matter how tiny, pokes out over the land. Shore points have no wiggle room.

Read the definition slowly, because its shape — “for every point there exists a radius” — will recur. The radius is allowed to depend on the point: points near the edge of $U$ may need very small balls. What is not allowed is a point where no radius at all works. Play with Figure 2.1.3: drag the point $x$ around different regions and watch the largest ball that fits.

Wiggle room Drag x (or focus it and use the arrow keys) · switch regions and rulers

Measure distance

Wiggle room. The largest straight-line ball around x that stays inside has radius $r \approx 0.58$ .

Open? no · Closed? yes — The rim belongs to the set, and rim points have no wiggle room.

Figure 2.1.3 A solid edge belongs to the set; a dashed edge does not. A set is open when every one of its points has a ball around it inside the set — so a solid edge anywhere spoils openness. Now switch the ruler: the balls change shape, but which points have room never changes.

Two facts about open sets are easy to check, and they will turn out to be the whole story.

Now for the observation that justifies the whole programme. Look again at the three unit balls in Figure 2.1.2: the diamond sits inside the disk, which sits inside the square. In general,

d_{\infty} (x, y) \leq d_{2} (x, y) \leq d_{1} (x, y) \leq 2 d_{\infty} (x, y) .

Read the middle inequality, $d_{2} \leq d_{1},$ as a statement about balls: anything within taxicab distance $r$ of $x$ is also within straight-line distance $r,$ so the diamond of radius $r$ sits inside the disk of radius $r .$ In the other direction, $d_{1} \leq 2 d_{2}$ (you will prove this in an exercise), so anything within straight-line distance $r / 2$ is within taxicab distance $r :$ a smaller disk sits inside every diamond. Either way, whenever a ball of one kind fits around a point, a ball of the other kind fits too.

Closed sets

What about sets like the closed disk, which contain their whole rim? They have their own name. The complement of a set $A \subseteq X$ is everything in $X$ that is not in $A;$ it is written $X ∖ A$ (read “ $X$ minus $A ”).$

The closed disk is closed because the part of the plane outside it is open: every point outside the disk has a little ball around it that stays outside. Intuitively, a set is closed when it contains its whole boundary — every point it gets arbitrarily close to. To say this precisely, call $x$ a limit point of $A$ if every ball around $x,$ however small, contains a point of $A$ other than $x$ itself. Then a set is closed exactly when it contains all of its limit points (an exercise asks you to see why). The closed interval $[0, 1]$ is closed; so is a single point; so is the set $Z$ of whole numbers.

The words “open” and “closed” come from these examples, and they are a little unfortunate, because in everyday language a door that is not open is closed. Sets are not like doors: as the epigraph of this chapter warns, a set can be open, or closed, or both, or neither. The half-open interval $[0, 1)$ is neither: it is not open (no room at $0)$ and not closed (it gets arbitrarily close to $1$ without containing it). And the whole real line $R$ is both: it is open (every point has all the room in the world) and closed (its complement is the empty set $\emptyset,$ which is open because it has no points that could fail the test). Sets that are both are sometimes called clopen.

Forgetting the ruler: topological spaces

We now have two kinds of information about the plane: distances, which stretching destroys, and open sets, which survive a change of ruler. Later in this chapter we will see that continuity, connectedness and compactness — all the concepts topology actually needs — can be described using open sets alone. So we make a bold move: we throw away the distance and keep only the list of open sets. To know which lists are allowed, we keep exactly the properties we proved above.

(The curly $T$ is just a capital T, for “topology”.) Notice what this definition does not say. It does not say what the points are: they can be numbers, pairs of numbers, letters, functions, anything. It does not mention distance, angles, straight lines or size. It describes a role — a list of sets that behaves like the open sets of a metric space — and anything that plays the role counts. The word “topology” is a little overloaded: “topology” is the subject, while “a topology on $X ”$ is a particular collection of open sets.

Why these rules?

The first answer is the one we have already given: these are exactly the properties that the open sets of every metric space have, so every metric space is automatically a topological space. Any theorem we prove about topological spaces will apply to all of them at once.

The second answer gives more insight into the strange asymmetry — any union, but only finite intersections. Think of an open set as a property you can confirm by a finite measurement. Suppose you want to know whether a number $x,$ which you can measure to any precision you like, lies in the open interval $(0, 1) .$ If it does, then a precise enough measurement will eventually show it: once you know $x$ to within a small enough error, you can be sure. (If $x$ is exactly $1,$ on the other hand, no finite measurement can ever confirm that $x < 1$ — which is why $(0, 1]$ is not open.) Now:

To confirm “ $x$ has property $P_{1}$ or $P_{2}$ or $P_{3}$ or …”, even infinitely many, it is enough to confirm one of them — a finite job. So unions of open sets are open.
To confirm “ $P_{1}$ and $P_{2} ”,$ you confirm both, one after the other — still a finite job. But to confirm infinitely many properties at once you might need infinitely many measurements, which you can never finish. So only finite intersections are guaranteed to be open.

This “observable properties” picture comes from computer science, where topology describes what a program can find out in finite time (Vickers 1989). Keep it alongside the wiggle-room picture; both will help later.

Examples, from blurry to sharp

Two extreme examples exist on every set, and they bracket all the others. In the discrete topology, every subset is open. It is the topology of the discrete metric (each single point is a ball of radius $\frac{1}{2},$ and any set is a union of single points), and it is the sharpest possible view: every point is isolated from every other. In the indiscrete topology, the only open sets are $\emptyset$ and $X .$ It is the blurriest possible view: no open set can tell any two points apart. The rules of a topology are satisfied in both cases — check!

Between the two extremes there are usually very many topologies. On a set with three points, $X = {a, b, c},$ there are exactly $29 .$ Figure 2.1.4 lets you hunt for them, and the checker tells you exactly which rule your collection breaks.

Build a topology Click a subset to make it open (or Enter on the keyboard)

$\emptyset$ and $X$ are open
unions of open sets are open
intersections of two open sets are open

A topology! This is a nested topology: each open set contains the smaller ones.

You have found 0 of the 29 topologies on three points.

Figure 2.1.4 The eight subsets of

X = {a, b, c},

each linked to the subsets one size bigger. Choose which ones are open. For instance

{\emptyset, {a}, {b}, X}

fails, because

{a} \cup {b} = {a, b}

is missing; add

{a, b}

and it becomes a topology.

A last piece of vocabulary. If $x$ is a point of a topological space, an open set containing $x$ is called a neighbourhood of $x .$ In a metric space, “ $U$ is a neighbourhood of $x ”$ means “ $U$ is open and contains a ball around $x ”$ — a region that gives $x$ some room.

New spaces from old: subspaces and products

The circle and the sphere are the most important spaces in this book, so we had better be able to say what their topologies are. They are not open subsets of anything; they are thin shapes sitting inside a bigger space. The rule is: an open set of the small space is what you see of an open set of the big space when you look only at the small space.

The unit circle $S^{1}$ is the set of points $(x_{1}, x_{2})$ in the plane with $x_{1}^{2} + x_{2}^{2} = 1 .$ An open disk in the plane that crosses the circle meets it in an open arc — an arc without its two end points — and these open arcs (and their unions) are the open sets of $S^{1} .$ In the same way the sphere $S^{2}$ sits in $R^{3},$ and its open sets are the open patches you cut out of it with open balls. The circle and the sphere are named after their dimension: $S^{1}$ is one-dimensional (an ant on it can only walk forwards or backwards) and $S^{2}$ is two-dimensional, a surface. The Warning box above was the same idea: $[0, \frac{1}{2})$ is open in the subspace $[0, 1]$ because it equals $(- \frac{1}{2}, \frac{1}{2}) \cap [0, 1] .$

There is a similar recipe for building a space from two others. Recall from §1.1 Sets and Functions that the Cartesian product $X \times Y$ is the set of all pairs $(x, y)$ with $x \in X$ and $y \in Y .$ The product topology on $X \times Y$ takes as its basic open sets the “open rectangles” $U \times V,$ with $U$ open in $X$ and $V$ open in $Y,$ and calls a set open when it is a union of such rectangles. The product $R \times R$ is the plane $R^{2}$ with its usual topology (an open disk is a union of tiny open rectangles). The product $S^{1} \times [0, 1]$ is a cylinder, and the product $S^{1} \times S^{1}$ of two circles is the surface of a doughnut — the torus, which we will meet again and again.

Continuity: no tearing

We have a precise notion of space. Now we need a precise notion of “stretching without tearing”. In topology, a stretching is a function — a rule that sends each point of one space to a point of another — with a special property called continuity. Let us discover the property on the real line first, where we can draw graphs.

Draw the graph of a function $f .$ Informally, $f$ is continuous if you can draw its graph without lifting your pen. Look at the function with a jump in Figure 2.1.5. Near $a = 2,$ the inputs are very close to each other, but the outputs leap from about $0.6$ to about $1.6 .$ Points that were neighbours have been torn apart. Continuity is the promise that this never happens: inputs that are close enough produce outputs that are close.

To make “close enough” precise, mathematicians play a game. A challenger names a tolerance $ε$ (the Greek letter epsilon, by tradition a small positive number). You must reply with a window $δ$ (delta) so that every input within $δ$ of $a$ produces an output within $ε$ of $f (a) .$ If you can win, however small the challenger makes $ε,$ the function is continuous at $a .$

The order of the quantifiers matters, exactly as in the game: first the challenger picks $ε$ (“for every $ε ”),$ then you answer with $δ$ (“there exists $δ ”),$ and your $δ$ may depend on the $ε$ you were given. Play the game below.

The ε–δ game Drag the teal ε, then the violet δ · drag a along the axis · press “smaller ε!”

Not this δ: part of the graph over the window escapes the band (shown in rose). No δ at all works for this ε: arbitrarily close to $a$ the graph jumps away. The function is not continuous at $a$ .

Figure 2.1.5 Teal: the band of outputs within

ε

f (a) .

Violet: the window of inputs within

δ

a .

You win if the graph over the window stays in the band (green). At the jump, once

ε

is smaller than half the jump, no window works, however narrow; the steep function needs very narrow windows, but some window always works.

From the game to open sets

The ε–δ definition speaks of distances, which we have promised to forget. Let us translate it. The inputs within $δ$ of $a$ form the ball $B (a, δ);$ the outputs within $ε$ of $f (a)$ form the ball $B (f (a), ε) .$ So continuity at $a$ says: for every ball around $f (a),$ some ball around $a$ is sent entirely inside it.

Now bring in the preimage from §1.1 Sets and Functions. For a function $f : X \to Y$ and a set $V \subseteq Y,$ the preimage $f^{- 1} (V)$ is the set of all points of $X$ that $f$ sends into $V :$

f^{- 1} (V) = {x \in X ∣ f (x) \in V} .

(Despite the notation, this makes sense even when $f$ has no inverse function: it simply asks “who lands in $V ?”)$ In this language, “ $B (a, δ)$ is sent inside $V ”$ means “ $B (a, δ) \subseteq f^{- 1} (V) ”.$ So a function is continuous exactly when, for every open $V$ in the target, every point of $f^{- 1} (V)$ has a little ball around it inside $f^{- 1} (V)$ — that is, when every preimage of an open set is open.

Condition (b) mentions only open sets, so it makes sense in any topological space. We take it as the definition.

Homeomorphisms: the same space, up to stretching

Film yourself stretching a rubber sheet, then play the film backwards: you see another perfectly good stretching, which puts every point back where it started. The wrapping of the line around the circle has no such reverse film: each point of the circle would have to go back to infinitely many places at once. Reversibility is exactly what “same shape” needs.

A bijection is a perfect matching: every point of $X$ is paired with exactly one point of $Y$ and vice versa, so nothing is glued. Continuity of $f$ means no tearing on the way there; continuity of $f^{- 1}$ means no tearing on the way back — which is the same as no gluing on the way there. A homeomorphism is a stretching you can perfectly undo.

Why insist that the inverse be continuous? Isn’t a continuous bijection enough? Figure 2.1.6 shows why not.

A continuous bijection that tears when reversed

Figure 2.1.6 Wrapping the half-open interval

[0, 1)

once around the circle is continuous and pairs points perfectly. But the small open arc

U

around

p

comes from two separate pieces at opposite ends of the interval. Run backwards, the map would rip

U

in two at

p :

its inverse is not continuous, and indeed

[0, 1)

and the circle are not homeomorphic.

And now, finally, the mug. A coffee mug — a solid one, made of clay — has a body and a handle with a hole through it. In Figure 2.1.7 the mug deforms continuously into a doughnut: first the dent that holds the coffee fills in (a dent is not a hole — it does not go all the way through), then the body shrinks into the handle while the handle thickens. At every moment the deformation is a perfect, reversible stretching. The gold loop threading the handle is a hint of what is to come: it can never be pulled free, not by any stretching, because the hole it goes around is never destroyed.

From coffee mug to doughnut Play or scrub the deformation · drag the picture to turn it

A coffee mug. Its dent is not a hole; its handle has one.

mug doughnut

Show the loop through the hole

Figure 2.1.7 A solid mug becomes a solid doughnut (a solid torus) without tearing or gluing, so the two are homeomorphic. The same deformation takes the mug’s surface to the doughnut’s surface, the torus. The gold loop around the handle survives the whole way: holes, unlike dents, cannot be stretched away.

Knots: one circle, many ways to sit in space

Our mug-to-doughnut film happens inside ordinary space, but a homeomorphism need not. Take a piece of string, tie a knot in it, and glue the two ends together. What you hold is a knot: a closed loop sitting in space. The simplest true knot, the trefoil, cannot be untangled into a round ring without cutting the string; anyone who has fought a tangled cable believes this, and §3.7 Knots and Links will prove it. Yet the trefoil and the circle are homeomorphic. Walk once around each and pair up the points in order: the pairing is continuous both ways, a perfect stretching. In Figure 2.1.8 every point of the knot wears the colour of its partner on the circle, and the colours run round the knot in the same order as round the wheel.

One circle, many knots Choose a knot · play or scrub to force it round · drag to turn it

Every point of the trefoil has the colour of the point of the circle it came from.

knot circle

Figure 2.1.8 A circle coloured like a wheel, and a knot whose points carry the colours of the points they came from: the knot is the circle, bent. The slider slides every point straight towards its place on a round circle. A trefoil or a figure-eight gets there only by passing through itself at one moment (the rose glow): a continuous movie of points may do that, a string may not. The tangled loop comes apart without it, because it was never knotted. The moment is found by computing a knot invariant, the Jones polynomial, all along the way.

So a homeomorphism compares spaces themselves, not the way they sit in the room: as a space, every knot is just a circle. What makes a knot a knot is how it is placed, and two knots count as the same knot only when one can be moved to the other through space without the string ever passing through itself, a much stricter sameness than homeomorphism. Telling knots apart is the business of knot theory, and it will call on almost every idea in this book. Loops take it up in §2.3 Homotopy, and it gets a chapter of its own in §3.7 Knots and Links.

Being homeomorphic behaves like a sameness should. Every space is homeomorphic to itself (use the identity); if $X ≅ Y$ then $Y ≅ X$ (use the inverse); and if $X ≅ Y$ and $Y ≅ Z$ then $X ≅ Z$ (compose). So homeomorphism is an equivalence relation in the sense of §1.2 Equivalence and Quotients, and it sorts all spaces into classes of “the same shape”.

A little history Rubber sheets and rulers

Johann Benedict Listing 1808 – 1882

Maurice Fréchet 1878 – 1973

Felix Hausdorff 1868 – 1942

The word Topologie first appears in an 1836 letter by Johann Listing, a student of Gauss, and in print in his book of 1847; a physicist by profession, Listing wrote a classic on the optics of the eye. Abstract distance came in Maurice Fréchet’s 1906 thesis (Fréchet 1906); in the First World War Fréchet spent two and a half years at or near the front, as an interpreter with the British Army. Felix Hausdorff, who also published poems and philosophy as “Paul Mongré”, gave Fréchet’s spaces the name “metric”, and his Grundzüge der Mengenlehre (1914) defined topological spaces by neighbourhoods. In January 1942, about to be interned by the Nazis, he took his own life with his wife and her sister. The open-set axioms in the form above won out over rivals in the following decades (Moore 2008) — so the definitions in this chapter are younger than the motor car.

How do you prove two spaces are different?

To show that two spaces are homeomorphic, one good example is enough: exhibit a homeomorphism. To show that they are not homeomorphic is a completely different kind of problem. There are infinitely many functions from one to the other, and we would have to rule out every single one. We cannot try them all. We need a cleverer idea.

The clever idea is the topological invariant: a property of spaces (or a number computed from them) that homeomorphic spaces always share. If $X$ has the property and $Y$ does not, then $X$ and $Y$ cannot be homeomorphic. This is the contrapositive at work, as in §0.2 How to Read Mathematics: “homeomorphic ⇒ same invariant” is equivalent to “different invariant ⇒ not homeomorphic”. Here are the first few invariants. Keep a scorecard as you read: at the end of the chapter we will set all of them on a sphere and a torus, and see which can tell the two apart.

Connectedness

The interval $[0, 1]$ is connected — this is really a deep fact about the real numbers, closely related to the intermediate value theorem. The space $[0, 1] \cup [2, 3]$ is not: the two intervals are open in it (subspace topology!) and split it in two. A space that is not connected falls apart into maximal connected pieces, its connected components, and the number of components is an invariant.

Why is connectedness invariant? Suppose $f : X \to Y$ is continuous and onto, and $Y$ splits into open pieces $U$ and $V .$ Then $f^{- 1} (U)$ and $f^{- 1} (V)$ are open (continuity!), disjoint, non-empty, and together they cover $X :$ so $X$ splits too. Taking the contrapositive: a continuous image of a connected space is connected. In particular a homeomorphism cannot connect what was split, or split what was connected.

There is a more hands-on version of the same idea. A path in $X$ from $x$ to $y$ is a continuous map $γ : [0, 1] \to X$ (gamma, a Greek letter often used for paths) with $γ (0) = x$ and $γ (1) = y$ — a journey through the space that starts at $x$ at time $0$ and arrives at $y$ at time $1 .$ A space is path-connected if any two of its points can be joined by a path. Every path-connected space is connected. The converse fails only for rather wild spaces (the standard example, the “topologist’s sine curve”, wiggles infinitely fast), and for every space we will meet in this book the two notions agree.

A space that is not path-connected falls apart into path components: two points lie in the same path component when some path joins them. (Being joined by a path is an equivalence relation — run one path and then the other — and the path components are its classes.) A homeomorphism carries paths to paths, so it matches up the path components of the two spaces one for one: the number of path components is an invariant too.

Compactness

The open interval $(0, 1)$ and the closed interval $[0, 1]$ look almost identical: they differ by two points. Are they homeomorphic? There is a feeling that $(0, 1)$ is “leaky”: you can walk towards $0$ forever, getting closer and closer, without ever arriving at a point of the space. The closed interval has no such escape routes. The property that captures this is called compactness. In the words of Evelyn Lamb, “Compact means small. It is a peculiar kind of small, but at its heart, compactness is a precise way of being small in the mathematical world” (Lamb 2017).

Here is the precise definition, which takes some getting used to. An open cover of $X$ is a collection of open sets whose union is all of $X$ — a way of covering the space with open patches, possibly infinitely many.

The interval $(0, 1)$ is not compact: the open intervals $(\frac{1}{2}, 1),$ $(\frac{1}{3}, 1),$ $(\frac{1}{4}, 1),$ … cover it (every point $x > 0$ lies in one of them), but any finitely many of them only cover $(\frac{1}{n}, 1)$ for the largest $n$ used, leaving out the points near $0 .$ The leak lets the cover escape. For subsets of $R^{n}$ there is a simple test, the Heine–Borel theorem: a subset of $R^{n}$ is compact exactly when it is closed and bounded (fits inside some big ball) (Munkres 2000, Thm 27.3). So $[0, 1],$ the circle, the sphere, the torus and the solid doughnut are compact; $(0, 1),$ $R$ and the open disk are not.

Compactness is an invariant, because a continuous image of a compact space is compact. (Pull the cover back with preimages, choose finitely many, and push forward again.) So $(0, 1) \neq ≅ [0, 1] .$ Notice that boundedness on its own is not an invariant: $(0, 1)$ is bounded and $R$ is not, and yet they are homeomorphic. What compactness detects is not size but the absence of escape routes — to infinity, or to a missing edge.

Cut points

Our sharpest tool so far is very simple. Remove a single point from a space and count the pieces that are left.

If $f : X \to Y$ is a homeomorphism, it restricts to a homeomorphism from $X ∖ {p}$ to $Y ∖ {f (p)} .$ So $f$ sends cut points to cut points, and a point whose removal leaves three pieces to a point whose removal leaves three pieces. This settles several questions at once:

A circle is not an interval. Removing any point from a circle leaves it in one piece, but removing a middle point from an interval cuts it in two.
A line is not a plane. Remove a point from $R :$ two pieces. Remove a point from $R^{2} :$ still one piece, because you can walk around the gap. So $R \neq ≅ R^{2},$ as you would hope.
$[0, 1)$ is not $(0, 1) .$ If $h$ were a homeomorphism, removing $0$ from $[0, 1)$ leaves one piece, but removing $h (0)$ from $(0, 1)$ leaves two.

The alphabet is a perfect playground. Think of each capital letter as a space — a shape made of thin strokes. Which letters are homeomorphic? Removing points tells them apart: an end of a stroke is never a cut point; a point in the middle of a stroke that lies on no loop cuts the letter in two; a junction where three strokes meet can cut it into three. In Figure 2.1.9, snip points out of letters and count.

The Letter Lab Pick a letter · click a point of it to remove that point

Click a point of the letter to remove it.

Try the junctions — and a point on a loop.

free ends: 2
3-way points: 2
4-way points: 0
loops: 1

Mark ends and branch points Sort the whole alphabet

Figure 2.1.9 Removing a junction of T leaves three pieces, but no single point of O leaves more than one. The numbers of free ends, of three-way and four-way junctions, and of loops are invariants, and in this font they sort the 26 capitals into eight classes. In other fonts the answer can change: if K’s arms met its stem at two different points, K would be like H instead of X.

One more property: Hausdorff

Some topologies are too blurry to be useful. In the indiscrete topology on two points, every neighbourhood of one point contains the other, so a sequence of points could “approach” both at once. Almost every space we care about has a property that rules this out.

Every metric space is Hausdorff: if $d (x, y) = r > 0,$ the balls of radius $r /2$ around $x$ and $y$ do not meet, by the triangle inequality. In a Hausdorff space a sequence of points can approach at most one limit. The property seems so obvious that you may wonder why we name it. The reason is that in the next chapter we will glue spaces together, and gluing carelessly can destroy it; and in §2.4 Manifolds and Surfaces it will be part of the very definition of a manifold.

Why we need more: holes

Cut points distinguish letters and lines, but they are blunt instruments. Remove a point from a sphere: one piece. Remove a point from a torus: one piece. Remove any finite number of points from either: still one piece. Both are connected, compact, Hausdorff, and without cut points. Every invariant in this chapter fails to tell them apart — and yet surely a sphere is not a doughnut?

The difference is a hole. On the torus there is a loop — the gold loop of Figure 2.1.7 — that cannot be shrunk away or pulled free, because it goes around the hole. On the sphere every loop can be shrunk to a point.

Here is a way to hunt for holes with nothing but this chapter’s tools. We cannot list all spaces, but we can map simple shapes into one and watch what happens. A map from a point picks out a point. A map from the segment $[0, 1]$ is a path, and asks which points can be joined. A map from a circle is a loop. And a map from the edge of a triangle asks the sharpest question: does it extend to the whole triangle? Can the loop be filled in? In the plane every such loop can be filled; on the torus the gold loop of Figure 2.1.7 cannot. David Farrell’s video series You Could Have Invented Homology builds up to exactly this test, drawing a triangle’s three-coloured edge in the plane and wrapped around a cylinder (Boarbarktree 2021).

Turning the test into a rigorous, computable invariant is what the rest of this book is about. In §2.2 Gluing Spaces Together we will learn to build tori, spheres and stranger surfaces from simple pieces; in §2.3 Homotopy we will make “shrinking a loop” precise. In Part III, homology will relax the test a little, accepting any surface whose only edge is the loop rather than insisting on a filled triangle (§3.5 Maps, Invariance, and First Triumphs shows a loop that passes the relaxed test and fails the strict one), and it will count holes of every dimension, with numbers that no homeomorphism can change.

Exercises

Exercise Open, closed, both or neither?

Classify each subset of $R$ as open, closed, both or neither: (a) $(0, 1);$ (b) $[0, 1];$ (c) $[0, 1);$ (d) $\emptyset;$ (e) $R;$ (f) ${0};$ (g) $Z;$ (h) the union of the intervals $(\frac{1}{n}, 1)$ for $n = 2, 3, 4, \dots;$ (i) the intersection of the intervals $(- \frac{1}{n}, \frac{1}{n})$ for $n = 1, 2, 3, \dots .$

Hint

For “closed”, look at the complement. For (h) and (i), work out what the union and the intersection actually are.

Solution

(a) open, not closed (it gets close to $0$ without containing it). (b) closed, not open ( $0$ has no room). (c) neither. (d) and (e) both. (f) closed (its complement $(- \infty, 0) \cup (0, \infty)$ is open) but not open. (g) closed (the complement is a union of open intervals $(n, n + 1))$ but not open. (h) the union is $(0, 1) :$ open, not closed. (i) the intersection is ${0} :$ closed, not open — an infinite intersection of open sets need not be open.

Exercise Two rulers, one topology

Show that $d_{2} (x, y) \leq d_{1} (x, y) \leq 2 d_{2} (x, y)$ for points of the plane. Deduce that every taxicab ball around $x$ contains a straight-line ball around $x,$ and vice versa, so the two metrics give the same open sets.

Hint

Write $a = ∣ x_{1} - y_{1} ∣$ and $b = ∣ x_{2} - y_{2} ∣ .$ Compare $(a + b)^{2}$ with $a^{2} + b^{2},$ and use $2 ab \leq a^{2} + b^{2},$ which holds because $(a - b)^{2} \geq 0 .$

Solution

With $a, b$ as in the hint, $d_{1} = a + b$ and $d_{2} = a^{2} + b^{2} .$ First, $(a + b)^{2} = a^{2} + 2 ab + b^{2} \geq a^{2} + b^{2},$ so $d_{1} \geq d_{2} .$ Second, $(a + b)^{2} = a^{2} + 2 ab + b^{2} \leq 2 (a^{2} + b^{2}),$ so $d_{1} \leq 2 d_{2} .$ Now if $d_{1} (y, x) < r$ then $d_{2} (y, x) < r :$ the taxicab ball $B_{1} (x, r)$ lies inside the round ball $B_{2} (x, r) .$ And if $d_{2} (y, x) < r / 2$ then $d_{1} (y, x) < r :$ the round ball $B_{2} (x, r / 2)$ lies inside $B_{1} (x, r) .$ So whenever a ball of one kind fits inside a set around $x,$ a smaller ball of the other kind fits too, and “open” means the same for both.

Exercise Closed sets keep their limit points

Let $A$ be a subset of a metric space $X .$ Recall that $x$ is a limit point of $A$ if every ball around $x$ contains a point of $A$ other than $x .$ Show that $A$ is closed exactly when it contains all of its limit points.

Hint

Both directions are about a point $x$ outside $A .$ If $A$ is closed, its complement is open: what does that say about the balls around $x ?$

Solution

Suppose $A$ is closed and $x \in / A .$ The complement $X ∖ A$ is open, so some ball $B (x, r)$ lies inside it. That ball contains no point of $A$ at all, so $x$ is not a limit point of $A .$ Hence every limit point of $A$ lies in $A .$

Conversely, suppose $A$ contains all of its limit points, and let $x \in / A .$ Then $x$ is not a limit point, so some ball $B (x, r)$ contains no point of $A$ other than $x;$ and $x$ itself is not in $A,$ so the ball misses $A$ completely and lies inside $X ∖ A .$ Every point of the complement has room around it, so $X ∖ A$ is open, and $A$ is closed.

Exercise Snip and count

Using cut points, show that (a) the letters T and X (in our font) are not homeomorphic; (b) the circle $S^{1}$ is not homeomorphic to the figure-eight (two circles touching at one point). Then (c) explain why the letters E and T are homeomorphic.

Solution

(a) Removing the junction of X leaves four pieces; no single point of T leaves more than three (its junction leaves exactly three, any other point at most two). A homeomorphism would have to send X’s junction to a point of T whose removal leaves four pieces, and there is none.

(b) Removing the touching point of the figure-eight leaves two pieces, but removing any point of a circle leaves one piece.

(c) E consists of one junction (on the stem, where the middle bar meets it) with three strokes coming out of it — up and along the top, out along the middle bar, down and along the bottom. The corners do not matter. So E is a “tripod”, exactly like T: stretch the three arms of one onto the three arms of the other.

Exercise Connectedness is an invariant

Prove carefully: if $f : X \to Y$ is continuous and $X$ is connected, then the image $f (X)$ is connected (with the subspace topology from $Y).$ Deduce that a connected space cannot be homeomorphic to a disconnected one.

Hint

Suppose $f (X)$ splits into two disjoint non-empty open pieces, and pull them back.

Solution

Suppose $f (X) = A \cup B$ with $A, B$ disjoint, non-empty and open in $f (X) .$ Then $A = U \cap f (X)$ and $B = V \cap f (X)$ for some open sets $U, V$ of $Y .$ Their preimages $f^{- 1} (U)$ and $f^{- 1} (V)$ are open in $X$ because $f$ is continuous. They cover $X$ (every $f (x)$ lies in $A$ or $B),$ they are disjoint (a point in both would map into $A \cap B = \emptyset),$ and both are non-empty ( $A$ and $B$ contain values of $f).$ That splits $X,$ contradicting connectedness. If $h : X \to Y$ were a homeomorphism with $X$ connected, then $Y = h (X)$ would be connected; so a connected space is never homeomorphic to a disconnected one.

Summary

Recap

A metric measures distance; balls $B (x, r)$ are the points less than $r$ away.
A set is open if each of its points has some wiggle room; closed if its complement is open. Sets can be open, closed, both or neither — and the answer depends on the surrounding space.
Different metrics can give the same open sets. A topology keeps only the open sets: $\emptyset$ and $X$ are open, unions are open, finite intersections are open.
A map is continuous when preimages of open sets are open — the ε–δ idea without distances, and reversed arrows once more. Continuous maps may glue but never tear.
A homeomorphism is a continuous bijection with continuous inverse. Homeomorphic spaces are “the same shape”: the mug and the doughnut, the interval and the line.
To prove spaces different, use invariants: connectedness, compactness, cut points, the Hausdorff property. They tell the letters of the alphabet apart, but not the sphere from the torus — for that we need holes.

Spaces and Continuity

Rubber-sheet geometry

Changes when you stretch

Survives any stretching

Measuring distance: metric spaces

Open sets: room to wiggle

Closed sets

Forgetting the ruler: topological spaces

Why these rules?

Examples, from blurry to sharp

New spaces from old: subspaces and products

Continuity: no tearing

From the game to open sets

Homeomorphisms: the same space, up to stretching

Knots: one circle, many ways to sit in space

How do you prove two spaces are different?

Connectedness

Compactness

Cut points

One more property: Hausdorff

Why we need more: holes

Exercises

Summary

Further reading

References