Foundations of Topology

Introduction to Topology What is Topology? Topology is the branch of mathematics that studies properties of objects that remain unchanged when you deform them continuously. Imagine stretching, twisting, or crumpling a rubber sheet—the sheet can change shape dramatically, but topology focuses on what stays the same through all these transformations. The key restriction is that these deformations must be continuous. You cannot tear the sheet, glue pieces together, create new holes, or fill in existing holes. These operations fundamentally change the object's topological character in ways that continuous deformations cannot. A topological space is the formal mathematical object: it's a set equipped with a topology, which is a structure that defines how we can continuously deform parts of that set. Any geometric space you're familiar with—like the plane, a sphere, or a three-dimensional room—can be given a topology. This is why topology provides a powerful, unified language for studying geometry. Topological Properties A topological property is any characteristic of a shape that survives all continuous deformations. If two objects share all topological properties, then in a deep sense, they are "the same" from a topological viewpoint, even if they look completely different. Three of the most important topological properties are: Dimension distinguishes objects that are fundamentally different in scale. A line is one-dimensional, a piece of paper (or any surface) is two-dimensional, and a ball is three-dimensional. No continuous deformation can turn a line into a surface or a surface into a solid—their dimensions are preserved. Compactness refers to whether an object is "closed and bounded." A circle is compact (it's a closed loop), but a line extending infinitely in both directions is not compact. This property is preserved under continuous deformations: you cannot continuously deform a circle into an infinite line. Connectedness describes whether an object is in one piece or multiple pieces. A single circle is connected, but two separate circles are not. Continuous deformations preserve connectedness: you cannot split a connected shape into multiple pieces without tearing it. Homeomorphism: The Central Concept Two spaces are homeomorphic when one can be continuously deformed into the other without cutting or gluing. More precisely, a homeomorphism is a bijective (one-to-one and onto) continuous function whose inverse is also continuous. The classic example is the coffee mug and the doughnut. These objects look completely different, but they are homeomorphic: A mug can be continuously deformed into a doughnut by gradually enlarging the handle and shrinking the cup until the handle becomes a hole. At no point do we tear or glue—the transformation is continuous throughout. This might seem surprising, but it makes sense topologically: both objects have exactly one hole. The mug's hole is in its handle; the doughnut's hole goes through its center. The shapes are different, but their fundamental topological structure is identical. Why does this matter? Homeomorphic spaces share all topological properties. If you prove something is true for a mug, it's automatically true for a doughnut. More importantly, homeomorphism tells us which objects are truly different topologically. A mug is not homeomorphic to a ball (a sphere), because a ball has no hole. You cannot continuously deform a ball into a mug without creating a hole—an operation forbidden in topology. Homotopy and Homotopy Equivalence While homeomorphism is the strongest notion of topological equivalence, homotopy provides a weaker but still powerful way to relate objects. A homotopy is a continuous deformation between two continuous functions. Think of it as a smooth animation from one function to another. More formally, if you have two continuous functions $f$ and $g$ from one space to another, a homotopy between them is a continuous family of functions that starts at $f$ and ends at $g$. Homotopy equivalence is a broader concept than homeomorphism. Two spaces are homotopy equivalent when each can be continuously deformed into a common larger space and then back. A key difference: homotopy equivalence can "ignore" holes in a controlled way. For example, a circle and a solid disk (the region inside a circle) are not homeomorphic—one has a hole and the other doesn't. However, they are homotopy equivalent because you can think of both as being continuously deformable within the plane. Homotopy equivalence preserves many important topological invariants (properties that don't change), making it useful even when homeomorphism is too restrictive. Continuity in Topology In calculus, you learned that a function is continuous if small changes in the input produce small changes in the output. Topology generalizes this concept using open sets. A function between topological spaces is continuous if the preimage of every open set is open. This is the formal definition that works in any topological space. Here's why this matches your calculus intuition: In the standard topology on the real numbers (derived from the distance metric), the open sets are exactly the intervals and unions of intervals. A function $f: \mathbb{R} \to \mathbb{R}$ is continuous in the topological sense if and only if it's continuous in the calculus sense. The definition is consistent! This topological definition of continuity is powerful because it works for abstract spaces where you don't have a notion of distance, only a notion of "which sets are open." Manifolds: Locally Euclidean Spaces A manifold is a topological space that locally looks like Euclidean space, even though it might look complicated globally. This is one of the most important structures in topology. An n-dimensional manifold is a space where every point has a neighborhood homeomorphic to n-dimensional Euclidean space ($\mathbb{R}^n$). This means if you zoom in closely enough to any point, the space looks like a flat n-dimensional space. One-dimensional manifolds include curves like lines and circles. However, a figure-eight is not a one-dimensional manifold, because at the point where the two loops meet, no neighborhood looks like a simple line—you'll see the two loops coming together. Two-dimensional manifolds are called surfaces. Examples include: The plane (flat, unbounded, non-compact) The sphere (curved, compact, no holes) The torus (shaped like a doughnut, compact, with one hole) Two surfaces that cannot be visualized in ordinary three-dimensional space without self-intersection are the Klein bottle and the real projective plane. These are legitimate two-dimensional manifolds—locally they look like a flat surface everywhere—but they have global properties that prevent them from being embedded smoothly in 3D space without crossing through themselves. The remarkable fact is that even though these exotic surfaces are hard to visualize, they are manifolds just like the sphere and torus. They satisfy the definition: at every point, you can find a neighborhood that looks like a flat plane.