Topology
Study of geometric properties preserved under continuous deformations.
Basic Concepts
Types of Topological Spaces
Exploring Different Spaces
Topology covers a variety of spaces, each with unique properties. The most common types are:
- Metric Spaces: These have a notion of distance, like the plane or 3D space.
- Discrete Spaces: Every point is isolated; think of a collection of separate dots.
- Continuous Spaces: Points are close together, such as a line or surface.
- Manifolds: Spaces that locally look like Euclidean space, such as the surface of a sphere.
How Are Spaces Used?
Mathematicians classify spaces to better understand their behaviors and relationships. For example, surfaces can be classified by the number of holes they have.
Fun Examples
- The surface of a donut (torus) is a manifold with one hole.
- The real number line is a classic example of a metric, continuous space.
Examples
A chessboard without the squares connected is a discrete space.
The shape of a drumhead is a manifold that looks flat locally but is round overall.
In a Nutshell
Topological spaces come in many forms, from simple lines to complex surfaces, each with special properties.