Topology: An Inquiry-Based Approach

A topological space $(X,\tau )$ is connected if $X$ cannot be written as the union of two disjoint, nonempty, open subsets. A subset $A$ of a topological space topological space $(X,\tau )$ is connected if $A$ is connected in the subspace topology.

The connected subsets of $\mathbb{R}$ are the intervals and the single point sets.

A subspace $C$ of a topological space $X$ is a connected component of $X$ if $C$ is connected and there is no larger connected subspace of $X$ that contains $C\text{.}$

One application of connectedness is the Intermediate Value Theorem that tells us that if $A$ is a connected subset of a topological space $X$ and if $f:A\to \mathbb{R}$ is a continuous function, then for any $a,b\in A$ and any $y\in \mathbb{R}$ between $f(a)$ and $f(b)\text{,}$ there is a point $x\in A$ such that $f(x)=y\text{.}$

A subset $S$ of a connected topological space $X$ is a cut set of $X$ if the set $X\setminus S$ is disconnected, while a point $p$ in $X$ is a cut point if $X\setminus \{p\}$ is disconnected. The property of being a cut set or a cut point is a topological invariant, so we can sometimes use cut sets and cut points to show that topological spaces are not homeomorphic.