7. Connectedness

Connectedness

Definition and Properties of Connected Spaces

Connected Space: A topological space $X$ is said to be connected if it cannot be divided into two disjoint nonempty open sets. In other words, there are no nontrivial clopen sets $sets that are both open and closed$ in $X$.

Properties of Connected Spaces:

• If $X$ is connected, then every continuous function from $X$ to the discrete two-point space $\{0,1\}$ is constant.
• If $X$ is connected and $A$ is a subset of $X$, then the closure of $A$ in $X$ is connected.

Path-Connectedness and Its Implications

Path-Connected Space: A topological space $X$ is said to be path-connected if for every pair of points $x,y$ in $X$, there exists a continuous function $\gamma :[0,1]\to X$ such that $\gamma (0)=x$ and $\gamma (1)=y$.

Implications of Path-Connectedness:

• Every path-connected space is connected, but the converse is not necessarily true.
• In a path-connected space, any two points can be joined by a path, implying a stronger form of connectedness.

Examples of Connected and Disconnected Spaces

Connected Space: The interval $[0,1]$ in the standard topology of $\mathbb{R}$ is connected. However, the union of two disjoint intervals, such as $[0,1]\cup [2,3]$, is disconnected.

Disconnected Space: The set of rational numbers $\mathbb{Q}$ in the standard topology of $\mathbb{R}$ is disconnected. This is because $\mathbb{Q}$ can be partitioned into two disjoint open sets: the set of irrational numbers and the set of rational numbers.