Connected set

In topology, a space $X$ is connected if it cannot be separated, that is there do not exist disjoint non-empty open sets $A,B$ such that $X = A \cup B$ (this is often expressed as $X = A \sqcup B$ ). For example, the set $(0,1) \cup (2,3)$ is not connected as a subspace of $\mathbb{R}$ .

There are several definitions that are related to connectedness:

$X$ is path-connected if for any two points $x, y \in X$ , there exists a continuous function $f: [0,1] \to X$ such that $f(0) = x, f(1) = y$ . Path-connectedness is a stronger condition that connectedness.
$X$ is locally connected at a point $x \in X$ if for any open set $U$ about $x$ , there exists a connected open set $V$ such that $x \in V \subset C$ . $X$ is locally connected if it is locally connected at every single point in $X$ . Perhaps surprisingly, local connectedness and connectedness are independent properties.
Similarly, $X$ is locally path-connected at a point $x \in X$ if for any open set $U$ about $x$ , there exists a path-connected open set $V$ such that $x \in V \subset C$ . $X$ is locally path-connected if it is locally path-connected at every single point in $X$ .

A space is totally disconnected if the only connected subspaces of $X$ are one-point sets. Examples of such a space include the discrete topology and the lower-limit topology.

Formation

One can build connected spaces using the following properties.

The image of a connected space under a continuous function is also connected.
The union of connected spaces that share a point in common is also connected.
The union of a connected space with its limit points is also connected.
The finite Cartesian product of a connected space is also connected (under the product topology).

Components

Connectedness can be used to define an equivalence relation on an arbitrary space $X$ . If we define equivalence relation $x \sim y$ if there exists a connected subspace of $X$ containing $x,y$ , then the resulting equivalence classes are called the components of $X$ . In a sense, the components are the maximally connected subsets of $X$ .

We can define path-components in the same manner.

The quasicomponents are the equivalence classes resulting from the equivalence relation $x \sim y$ if there does not exist a separation $X = A \sqcup B$ such that $x \in A, y \in B$ . This definition is weaker than that of a component, for any component must lie in a quasicomponent (the definitions are equivalent if $X$ is locally connected).

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Connected set

Formation

Components