 TOPICS   # Topics in a Topology Course

To learn more about a topic listed below, click the topic name to go to the corresponding MathWorld classroom page.

### General

 Differential Topology Differential topology is the mathematical study of smooth manifolds. Dimension Dimension is a topological measure of the size of an object's covering properties, roughly corresponding to the number of coordinates needed to specify a point on the object. Homology Homology is a mathematical concept used in many branches of algebra and topology that involves a topological invariant known as a homology group. Homotopy A continuous deformation of a topological space or a function between two topological spaces. Knot A closed, nonself-intersecting curve that is embedded in three dimensions and that cannot be untangled to produce a simple loop. Link A link is an assembly of knots with mutual entanglements. Manifold A manifold is a topological space that is locally Euclidean, i.e., around every point, there is a neighborhood that is topologically the same as an open unit ball in some dimension. Metric A metric is a nonnegative function describing the distance between neighboring points for a given set. Metric Space A metric space is a set with a global distance function that for every two of the set's points gives the distance between them as a nonnegative real number. Möbius Strip A Moebius strip is one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving it half a twist, and then reattaching the two ends. Projective Plane The projective plane is the set of lines in the Euclidean plane that pass through the origin. It can also be viewed as the Euclidean plane together with a line at infinity. Projective Space Projective space is the generalization of the projective plane to more than two dimensions. Tangent Space A tangent space is a vector space of all possible tangent vectors to a point on a manifold. Topology (1) As a branch of mathematics, topology is the mathematical study of object's properties that are preserved through deformations, twistings, and stretchings. (2) As a set, a topology is a set along with a collection of subsets that satisfy several defining properties. Torus A torus is a closed surface containing a single hole that is shaped like a doughnut. Vector Bundle Given a topological space X, a vector bundle is a way of associating a vector space to each point of X in a consistent way.

### Point-Set Topology

 Closed Set: A closed set is a subset of a topological space that contains all of its limit points. A closed interval is an example of a closed set. Homeomorphism: A homeomorphism is ann equivalence relation and one-to-one correspondence that is continuous in both directions between points in two geometric figures or topological spaces. Neighborhood: The neighborhood of a point is an open set containing that point. Open Set: An open set is a set for which every point in the set has a neighborhood lying in the set. An open set is the complement of a closed set and. An open interval is an example of an open set. Point-Set Topology: Point-set topology is the study of the general abstract nature of continuity on spaces. Basic point-set topological notions are ones like continuity, dimension, compactness, and connectedness. Subspace: A subspace is a subset of a vector space that is also itself a vector space. This term can also be used for a subset of a topological space. Topological Space: A topological space is a set with a collection of subsets T that together satisfy a certain set of axioms defining the topology of that set.