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  "body": "\n## Definition\n\nLet $X \\subseteq \\mathbb{R}^n$.\nThe set $X$ is simply connected if $X$ is path-connected and each continuous [[closed curve|0D032NSZKY]] $c \\colon [0, 1] \\to X$ 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\n$$H(t, 0) = c(t), \\qquad H(t, 1) = p, \\qquad H(0, s) = H(1, s) \\quad \\text{for all } s, t \\in [0, 1].$$\nAn open simply connected set is a simply connected domain.\n\n## Remarks\n\nThe map $H$ deforms the loop $c$ into the constant loop at $p$ through closed loops $t \\mapsto H(t, s)$ in $X$.\nEach convex set is simply connected. For example, $\\mathbb{R}^n$ and each ball are simply connected.\nThe 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 $n \\ge 3$.\nAn 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.\n\n## Explanations\n\n### L1 human\n\nA set is simply connected when every loop in it can be pulled tight to a point, like a rubber band, without leaving the set.\nA disc has this property. An annulus does not, because a loop around the hole cannot shrink past the hole.\n",
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