Simply connected domain

Definition

Let $X \subseteq \mathbb{R}^n$. The set $X$ is simply connected if $X$ is path-connected and each continuous closed curve $c \colon [0, 1] \to X$ has this property. There are a point $p \in X$ and a continuous map $H \colon [0, 1] \times [0, 1] \to X$ such that $$H(t, 0) = c(t), \qquad H(t, 1) = p, \qquad H(0, s) = H(1, s) \quad \text{for all } s, t \in [0, 1].$$ An open simply connected set is a simply connected domain.

Remarks

The map $H$ deforms the loop $c$ into the constant loop at $p$ through closed loops $t \mapsto H(t, s)$ in $X$. Each convex set is simply connected. For example, $\mathbb{R}^n$ and each ball are simply connected. The plane minus the origin is not simply connected, and neither is $\mathbb{R}^3$ minus the $z$-axis. But $\mathbb{R}^n$ minus a point is simply connected for $n \ge 3$. An equivalent condition: $X$ is path-connected, and any two paths in $X$ with the same end points can be deformed into each other with the end points fixed.

Explanations

L1 human

A set is simply connected when every loop in it can be pulled tight to a point, like a rubber band, without leaving the set. A disc has this property. An annulus does not, because a loop around the hole cannot shrink past the hole.