Tag Archives: pentagon

Confusing terms in topology

Standard

Following are some of the terms used in topology which have similar definition or literal English meanings:

  • Convex set: A subset U of \mathbb{R}^n is called convex1 , if it contains, along with any pair of its points x,y, also the entire line segement joining the points.
  • Star-convex set: A subset U of \mathbb{R}^n is called star-convex if there exists a point p\in U such that for each x\in U, the line segment joining x to p lies in U.
  • Simply connected: A topological space X is called simply connected if it is path-connected2  and any loop in X defined by f : S^1 \to X can be contracted3  to a point.
  • Deformation retract: Let U be a subspace of X. We say  is a X deformation retracts to U if there exists a retraction4 r : X \to U a retraction such that its composition with the inclusion is homotopic5  to the identity map on X.
convex

Various examples to illustrate the interdependence of these terms. Shown here are pentagon, star, sphere, and annulus.

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 S^n is simply connected if and only if n \geq 2. Every star-convex subset of \mathbb{R}^n 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 S^n, that is, the (n-1)-sphere, is not a retract of the ball. Using this we can prove the Brouwer fixed-point theorem. However, \mathbb{R}^n-0 deformation retracts to a sphere S^{n-1}. Hence, though the sphere shown above doesn’t deformation retract to a point, it is a deformation retraction of \mathbb{R}^3-0.


Footnotes

  1. 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 a and b are points in the vector space, the points on the straight line between a and b are given by x = \lambda a + (1-\lambda)b for all \lambda from 0 to 1.
  2. A path from a point x to a point y in a topological space X is a continuous function f from the unit interval [0,1] to X with f(0) = x and f(1) = y. The space X is said to be path-connected if there is a path joining any two points in X.
  3. There exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. In general, a space X is contractible if it has the homotopy-type of a point. Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations.
  4. Then a continuous map r: X\to U is a retraction if the restriction of r to U is the identity map on U.
  5. A homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H : X \times [0,1] \to Y such that, if x \in X then H(x,0) = f(x) and H(x,1) = g(x). 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.
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