Following are some of the terms used in topology which have similar definition or literal English meanings:
- Convex set: A subset of is called convex1 , if it contains, along with any pair of its points , also the entire line segement joining the points.
- Star-convex set: A subset of is called star-convex if there exists a point such that for each , the line segment joining to lies in .
- Simply connected: A topological space is called simply connected if it is path-connected2 and any loop in defined by can be contracted3 to a point.
- Deformation retract: Let be a subspace of . We say is a deformation retracts to if there exists a retraction4 a retraction such that its composition with the inclusion is homotopic5 to the identity map on .
A stronger version of Jordan Curve Theorem, known as Jordan–Schoenflies theorem, implies that the interior of a simple polygon is always a simply-connected subset of the Euclidean plane. This statement becomes false in higher dimensions.
The n-dimensional sphere is simply connected if and only if . Every star-convex subset of is simply connected. A torus, the (elliptic) cylinder, the Möbius strip, the projective plane and the Klein bottle are NOT simply connected.
The boundary of the n-dimensional ball , that is, the -sphere, is not a retract of the ball. Using this we can prove the Brouwer fixed-point theorem. However, deformation retracts to a sphere . Hence, though the sphere shown above doesn’t deformation retract to a point, it is a deformation retraction of .
- In general, a convex set is defined for vector spaces. It’s the set of elements from the vector space such that all the points on the straight line line between any two points of the set are also contained in the set. If and are points in the vector space, the points on the straight line between and are given by for all from 0 to 1.
- A path from a point to a point in a topological space is a continuous function from the unit interval to with and . The space is said to be path-connected if there is a path joining any two points in .
- There exists a continuous map such that restricted to is . Here, and denotes the unit circle and closed unit disk in the Euclidean plane respectively. In general, a space is contractible if it has the homotopy-type of a point. Intuitively, two spaces and are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations.
- Then a continuous map is a retraction if the restriction of to is the identity map on .
- A homotopy between two continuous functions and from a topological space to a topological space is defined to be a continuous function such that, if then and . Deformation retraction is a special type of homotopy equivalence, i.e. a deformation retraction is a mapping which captures the idea of continuously shrinking a space into a subspace.