Connectedness (mathematics)
Connectedness in mathematics refers to a fundamental property of topological spaces, where a space is considered connected if it appears as a single piece without any separations. Formally, a topological space is defined as connected if it cannot be divided into two non-empty open sets that do not intersect. This concept is invariant under homeomorphisms, meaning that a connected space retains its connectedness even when smoothly transformed. Conversely, spaces that consist of multiple connected components are labeled as disconnected.
Connectedness can take various forms, such as path connectedness, where any two points within a space can be connected by a continuous path that remains entirely within that space. While all path-connected spaces are connected, the reverse isn't necessarily true; there are connected spaces that are not path connected. In algebraic topology, further distinctions are made, such as simply connected spaces, like a solid disk, where paths between points can be continuously deformed into one another, versus multiply connected spaces, like an annulus, where distinct classes of paths exist due to a central hole. Understanding these concepts is vital for exploring more complex topological and geometric structures in mathematics.
Subject Terms
Connectedness (mathematics)
In mathematics, connectedness is an intrinsic property of a topological space. Intuitively, a topological space is connected if it is all one piece. The connectedness of a topological space is invariant in the sense that it is preserved under homeomorphism; that is, a connected space will remain connected under a smooth deformation. A topological space made up of more than one connected component is disconnected. The notion of connectedness appears in various forms across mathematics, typically because the mathematical object in question would be connected if considered as a topological space.
Topological Connectedness
A topological space consists of a set X, together with a collection T of subsets of X, such that both the empty set ∅ and the set X itself are elements of T; any union (finite or infinite) of elements of T is also in T; and any finite intersection of elements of T is also in T. Then T is called a topology on X, and the elements of T are called open sets.
A topological space is connected if it cannot be expressed as the union of two disjoint nonempty open sets.
In the Euclidean plane ℝ2, an open ball is the set of all points in the interior of a circle. A closed ball, on the other hand, includes the boundary of the circle as well. In the standard topology on the Euclidean plane, a set A is an open set if and only if each point of A lies within an open ball that is itself contained within A.
This induces a topology on any subset S of the Euclidean plane; the open subsets of S are the intersection of S with the open subsets in the standard topology on ℝ2.
In some cases, a connected space may become disconnected by the removal of one point, as with a simple curve in the plane. A circle may be disconnected by removing two points, while a solid disk in the plane may be disconnected by removing points along a line or other one-dimensional curve.
Other Types of Connectedness
A topological space is said to be path connected when any two points in the space can be joined by a continuous path contained entirely within the space. Any path-connected topological space is also connected; however, there are instances of connected topological spaces that are not path connected.
In algebraic topology, path-connected spaces are further categorized by the way in which they are connected.
A solid disk is a simply connected topological space. Given two points in a disk, any path joining a given pair points can be continuously deformed, without breaking, to any other such path. That is, all paths are equivalent up to homeomorphism.
An annulus, on the other hand, is a multiply connected topological space because it is not simply connected. Given two points in an annulus, there are two distinct classes of paths, passing to either side of the hole in the center. These paths are not homeomorphic because one cannot be deformed across the hole to the other without introducing a break in the path.
Bibliography
Insall, Matt and Eric W. Weisstein. "Connected Set." Wolfram MathWorld. Wolfram Research Inc. Web. 11 Mar. 2016. http://mathworld.wolfram.com/ConnectedSet.html
Munkres, James R. Topology. 2nd ed. Upper Saddle River: Prentice Hall, 2000. Print.