Path-connected space

Definition

Let $X$ be a topological space. A path in $X$ from $x$ to $y$ is a continuous map $\gamma \colon [0, 1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$. The space $X$ is path-connected when $X$ is nonempty and for all $x, y \in X$ there is a path in $X$ from $x$ to $y$. A subset of $X$ is path-connected when it is path-connected in the subspace topology.

Remarks

Some authors do not require $X$ to be nonempty. Each convex subset of $\mathbb{R}^n$ is path-connected: use the segment $\gamma(t) = (1 - t)x + ty$. Path-connected spaces are connected. The topologist's sine curve is connected but not path-connected.

Explanations

L1 human

A space is path-connected when you can walk from each point to each other point along a continuous path inside the space. This is a concrete way to show that a space is in one piece.