Topology is the study of properties that survive deformation. Stretch a donut into a coffee mug — topologically, they're the same. The hole is what matters, not the shape. This sounds like a game, but it turns out to be enormously powerful: properties that survive deformation tell you something about deep structure rather than surface appearance.
Consider Brouwer's fixed point theorem: any continuous map of a disk to itself has at least one point that doesn't move. No matter how you stir a cup of coffee, some point in the liquid returns exactly to where it started. This is not a quantitative result — it doesn't tell you where the fixed point is — but it guarantees existence. Topology often works this way: it tells you what must be true without telling you where.
Or consider the hairy ball theorem: you cannot comb the hair on a sphere smooth without a cowlick. There must be at least one point where the hair stands straight up or disappears. This applies to Earth's atmosphere: there must always be at least one point on Earth's surface where the horizontal wind speed is zero. The topology of the sphere forces it.
What I find most interesting about these results is how they differ from the usual style of mathematical argument. Most mathematics tells you what is true and derives it from axioms. Topology tells you what must be true given the shape of the space — before you know any of the details. The constraints come from the global structure, not the local rules.
I think this is a useful mode of reasoning more generally. Some constraints are topological: they follow from the shape of the situation, not from the specific values. The existence of a fixed point isn't a fact about how you stirred the coffee. It's a fact about the topology of the cup.
→ global order from local rules — Penrose tiling