Connected Space · aiwiki.page

A connected space is a topological space that cannot be divided into two disjoint, nonempty open subsets. Connectedness formalizes the idea that a space consists of one piece, but does not necessarily mean that any two points can be joined by a continuous path. It is a fundamental property in topology and is preserved by homeomorphisms. (math.toronto.edu)

Definition and equivalent characterizations

A separation of a topological space $X$ is a pair of open sets $U,V\subseteq X$ satisfying

X=U\cup V,\qquad U\cap V=\varnothing,\qquad U\ne\varnothing,\quad V\ne\varnothing.

The space is connected if no separation exists, and disconnected otherwise. In a separation, each set is also closed, because its complement is the other open set. Thus connectedness can equivalently be defined by the impossibility of partitioning $X$ into two nonempty, disjoint closed sets. (math.toronto.edu)

A set that is both open and closed is called clopen. The equivalent clopen characterization is:

X\text{ is connected} \quad\Longleftrightarrow\quad \text{the only clopen subsets of }X\text{ are }\varnothing\text{ and }X.

Under the no-separation definition, the empty space and every one-point space are connected. (math.toronto.edu)

A subset $A\subseteq X$ is called connected when it is connected with its subspace topology. Consequently, a separation of $A$ uses sets open in $A$ ; those sets need not be open in the ambient space $X$ . Connectedness depends on the topology, not merely on the underlying set. For example, any set with the indiscrete topology is connected, whereas a set with the discrete topology is disconnected whenever it contains at least two points. (math.toronto.edu)

Connectedness on the real line

A nonempty subset of the real line with its usual topology is connected exactly when it is an interval, allowing one-point intervals and unbounded intervals. Equivalently, if it contains $a<b$ , it must contain every number between them. If an intermediate point $c$ is missing, intersection with the two open half-lines determined by $c$ gives a separation. (math.toronto.edu)

This characterization explains the intermediate value theorem topologically. If $X$ is connected and $f:X\to\mathbb R$ is continuous, then for any $a,b\in X$ , every value between $f(a)$ and $f(b)$ is attained somewhere in $X$ . The essential fact is that the image of a connected space under a continuous function is connected, and therefore is an interval when it lies in $\mathbb R$ . (math.toronto.edu)

Preservation and construction

Several basic results allow connected spaces to be constructed and recognized:

Continuous images: If $X$ is connected and $f:X\to Y$ is continuous, then $f(X)$ is connected. A separation of the image would pull back to a separation of $X$ . In particular, connectedness is invariant under homeomorphisms. (math.toronto.edu)
Closure: If $A\subseteq X$ is connected, then its closure $\overline A$ is connected. More generally, every set $B$ satisfying $A\subseteq B\subseteq\overline A$ is connected. (math.toronto.edu)
Unions with overlap: A family of connected subsets with a common point has a connected union. More generally, it suffices that one connected member intersect every other member. (math.toronto.edu)
Products: A product of connected spaces is connected when equipped with the product topology, including an infinite product. The topology specified here is essential to the theorem. (math.toronto.edu)

Connectedness is not inherited by arbitrary subsets. For example, the interval $[0,1]$ is connected, but its two-point subset $\{0,1\}$ is not. (math.toronto.edu)

Path-connectedness

A space is path-connected if, for every $x,y\in X$ , there exists a continuous map

\gamma:[0,1]\longrightarrow X, \qquad \gamma(0)=x,\quad \gamma(1)=y.

Such a map is called a path. Every path-connected space is connected, because paths are continuous images of the connected interval $[0,1]$ . The converse fails for general topological spaces. (math.toronto.edu)

A standard counterexample is the closed topologist’s sine curve:

S= \left\{\left(x,\sin\frac1x\right):0<x\leq1\right\} \cup \bigl(\{0\}\times[-1,1]\bigr),

with the topology inherited from the plane. The oscillating graph is path-connected, and $S$ is its closure, so $S$ is connected. Nevertheless, no path in $S$ joins the vertical segment to the graph: the oscillations near $x=0$ prevent a continuous transition. Thus connectedness does not imply the existence of paths between points. (math.toronto.edu)

Connected components

The connected component containing a point $x\in X$ is the largest connected subset containing $x$ . It can be expressed as

C(x)= \bigcup\{A\subseteq X:A\text{ is connected and }x\in A\}.

The union is connected because all its members contain $x$ . Components are therefore maximal connected subsets, and every space is partitioned into its connected components. (math.mit.edu)

Every connected component is closed, but it need not be open. A nonempty space is connected precisely when it has one connected component. Path components are defined instead by the existence of paths between points; each path component lies within a connected component, but the two types need not coincide. The closed topologist’s sine curve has one connected component and two path components. (math.toronto.edu)

A space is totally disconnected if every connected component is a singleton. This does not imply that its topology is discrete. For example, the rational numbers with their usual subspace topology are totally disconnected, although no rational point is isolated. More generally, every countable metric space is totally disconnected: around a point, one can choose arbitrarily small ball radii that are not distances to any other point, making those balls clopen. (people.math.harvard.edu)

Local connectedness

A space is locally connected if every neighborhood of each point contains a connected open neighborhood of that point. It is locally path-connected if the corresponding neighborhoods can be chosen path-connected. Equivalently, these conditions require bases of connected or path-connected open sets. (math.toronto.edu)

These are local conditions, distinct from global connectedness. Two disjoint open intervals form a locally path-connected but disconnected space. Conversely, the closed topologist’s sine curve is connected but not locally connected at points of its vertical segment. In a locally connected space, connected components are open as well as closed. (math.mit.edu)

In a locally path-connected space, connected components and path components coincide. Hence such a space is connected if and only if it is path-connected. An important consequence is that every connected open subset of Euclidean space is path-connected; openness supplies small path-connected neighborhoods. (math.toronto.edu)

Distinguishing spaces

Connectedness can distinguish spaces that are not homeomorphic. The number of connected components is a topological invariant. A more refined invariant is the behavior under deletion of points: a cut point of a connected space is a point whose removal disconnects it. Every point of the real line is a cut point, whereas no point of the Euclidean plane is one, so the line and plane cannot be homeomorphic. (math.toronto.edu)

References

MAT327: Introduction to Topology — Lecture 9math.toronto.edu
MAT327: Introduction to Topology — Lecture 10math.toronto.edu
MAT327: Introduction to Topology — Lecture 11math.toronto.edu
MAT246: Outline of Point Set Topologymath.toronto.edu
Topology Notes — Math 131, Harvard Universitypeople.math.harvard.edu
MAT327: Introduction to Topology — Assignment 3math.toronto.edu
Topology: A First Course — Local Connectednessmath.mit.edu