5. Topology

Open sets, compactness, completeness — the topology of the real line, the Baire category theorem, Cantor sets.

• Open and Closed Sets — This scene shows what an open set is. Move the point and the radius, and check whether the small ball fits inside. Open means no point clings to the boundary. Around every point there is a small interval lying inside the set. At a boundary point, every interval, however small, pokes outside. For example, kick a ⚽ ball in the middle of a yard. A gentle kick keeps it inside. Kick from right at the fence, and it flies out at once. That is the difference between an interior point and a boundary point.
• Clopen Sets — This scene is about sets both open and closed: clopen sets. The same set changes its fate with the space it lives in. On the whole real line, only the empty set and the line itself qualify. A boundary always appears somewhere. But if the space splits into two separate pieces, one piece touches no boundary. That piece is both open and closed. For example, take one of two far-apart 🏠 houses. In a world made of that house alone, it forms its own separate region. It is open, and it is closed.
• Connected Sets — This scene is about connected sets. Move the cut point and check whether the set splits into two open pieces that never touch. A set that cannot be split is connected. An interval never splits: wherever you cut, the cut point itself belongs to neither side. Two far-apart pieces, or the rationals with their dense gaps, split cleanly when the cut lands in a gap. For example, look at one 🧩 jigsaw puzzle. While the pieces interlock, it is a single board. Lying in two completely separate piles, it is already split apart.
• Closure and Boundedness — This scene builds the closure of a set. Watch what happens when every escaped limit point is pulled back in. The closure contains every point the set was reaching for. If some point is approached but never touched, closing means adding it. A set already holding all its limit points equals its closure. Add boundedness, and compactness is near. For example, fill a 📦 box and put the lid on. Only with the lid is the box complete. Closure is like that lid: it adds the boundary points to the set.
• Compact Sets — This scene tests compactness with open covers. See whether finitely many covering pieces always suffice. Compact means closed and bounded together. An open end needs endless pieces near the gap. An endless stretch fails the same way. With both conditions, any cover shrinks to finitely many pieces. For example, cover a desk completely with 📚 books. A finite desk needs only a few books. An endless desk would defeat any pile, however large.
• Topologist's Sine Curve — This scene draws the topologist's sine curve. Watch it oscillate endlessly near zero, and the limit segment it sticks to. The closer to zero, the faster the curve wiggles. So a whole segment on the axis sticks to the curve. The combined figure is in one piece. Yet no path walks from the segment into the curve. Being in one piece and having a path are different things. For example, shake one end of a 🧵 string faster and faster. The shaking end blurs into a band. The band is visible, but you cannot walk from one point of it onto a chosen part of the string.
• Peano Curve — This scene builds the Peano curve, one line filling a square. Watch the approximations grow denser at every stage. The curve folds like a snake, filling the gaps. More folds mean more points touched. In the limit it passes through every point of the square. But it must visit some points twice. So no injective filling exists. For example, color a whole sheet with a ✏️ pencil, never lifting it. Zigzag densely back and forth, and eventually every point is touched. One line fills an entire surface.
• Metric Spaces — This scene axiomatizes distance. Compare how the shape of a ball changes when the metric changes. A metric obeys only three rules. Distances are never negative. The distance is the same both ways. A detour is never shorter than the direct way. Same rules, different measurement: balls of equal radius take different shapes. For example, change how a 📏 ruler measures between two points. Measure straight, and you get a disk. Measure along grid lines only, and you get a diamond. Same radius, different reach.
• Contraction Mapping Principle — This scene runs the contraction mapping principle. Iterate and watch the points tighten onto one fixed point. A contraction shrinks every distance by one fixed ratio. Iterating pulls all points toward each other. Every start leads to the same destination. But if the space has a hole, the destination can fall into it. For example, fold a 🗺️ map in half, then in half again. Each fold shrinks the paper, and all points gather toward one. That point is the fixed point. Tear a hole in the middle, and the gathering point is gone.
• Baire Category Theorem — This scene follows the Baire category theorem through nested intervals. Dodge the rationals one by one, and a point still remains. Sets full of holes cannot cover everything. Choose smaller and smaller closed intervals, each dodging the next bad point. The intervals tighten onto one point belonging to no hole set. A complete space always keeps such a point. For example, put a 📦 box inside a box, then a smaller one inside that. However small it gets, room remains at the center. Choosing the boxes to dodge rationals still leaves one point standing.
• Baire Class 1 Functions — This scene builds a Baire class 1 function. Thomae's function rises as a pointwise limit of continuous tent sums. Stacking only continuous functions can still produce breaks. But the breaks cannot be everywhere. Points of continuity are always dense. Even a limit-made function keeps this much order. For example, bend a 🧵 string into ever sharper mountain shapes. At every stage the string is unbroken. In the end, a row of thorn-like spikes remains.
• Length Zero, Points Uncountable — This scene builds the Cantor set. Keep deleting middle thirds: the length dies to zero, yet uncountably many points survive. Each deletion shrinks the total length to two-thirds. After infinitely many rounds, the length is zero. But some points are never cut away. There are as many of them as there are real numbers. For example, cut the middle third out of a 🍰 cake. Then cut the middle of each remaining piece, again and again. The cake almost vanishes. Still, uncountably many spots were never touched by the knife.
• The Fat Cantor Set — This scene builds the fat Cantor set. Dig out only a little at each stage, and half the length survives. Digging less changes the outcome. The remaining set is still full of holes, with empty interior. Yet its length converges to one half. Size by length and size by topology are independent. For example, scoop a 🍰 cake again, but take only small bites. Holes appear everywhere, yet half the cake remains to eat. It looks empty, but the amount is real.
• The Cantor Function — This scene climbs the Cantor function, the Devil's staircase. It is continuous and always rising, yet flat on almost every interval. The function pauses on every gap of the Cantor set. So its slope is zero at almost every point. And still it climbs from 0 to 1 overall. The total climb disagrees with the sum of slopes. For example, climb a 🪜 staircase whose landings are endlessly long. You rest almost the whole way, yet you still reach the top. The Cantor function does exactly that.