Topology: An Inquiry-Based Approach

A path in a topological space $X$ is a continuous function $p$ from the interval $[0,1]$ to $X\text{.}$ If $p(0)=a$ and $p(1)=b\text{,}$ then $p$ is a path from $a$ to $b\text{.}$

A subspace $A$ of a topological space $X$ is path connected if, given any $a,b\in A$ there is a path in $A$ from $a$ to $b\text{.}$

The path component of an element $a$ in a topological space $(X,\tau )$ is the largest path connected subset of $X$ that contains $a\text{.}$

A topological space $(X,\tau )$ is locally path connected at $x$ if every neighborhood of $x$ contains a path connected subset with $x$ as an element. The space $(X,\tau )$ is locally path connected if $X$ is locally path connected at every point.

Connectedness and path connectedness are equivalent in finite topological spaces, and path connectedness implies connectedness in general. However, there are topological spaces that are connected but not path connected. One example is the topologist’s sine curve.