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  "body": "\n## Example\n\nLet $Q = [0, 1] \\times [0, 1]$ be the unit square.\nThe *Klein bottle* $K$ is the quotient of $Q$ by the identifications\n$$(0, y) \\sim (1, y) \\quad \\text{and} \\quad (x, 0) \\sim (1 - x, 1) \\qquad \\text{for } x, y \\in [0, 1].$$\nThen $K$ is a closed [[surface|VGJJVBZ1MQ]].\nCheck: every point has a disc neighbourhood, made of pieces of $Q$ glued along the identified edges.\nAlso $K$ is compact and connected as a quotient of $Q$.\nThe four corners become one vertex, the edges give two edges, and $Q$ gives one face, so the Euler characteristic is $\\chi(K) = 1 - 2 + 1 = 0$.\nThe strip $\\set{(x, y) : \\abs{x - 1/2} \\le 1/4}$ with its top and bottom edges glued by $(x, 0) \\sim (1 - x, 1)$ is a [[Möbius strip|04AJE0M271]] in $K$, so $K$ is not orientable.\n\n## Remarks\n\nBy the [[classification of surfaces|2SFM73YET0]], $K$ is homeomorphic to the connected sum $\\R P^2 \\# \\R P^2$.\nIts fundamental group has the presentation $\\langle a, b \\mid a b a^{-1} b \\rangle$.\nGluing two Möbius strips along their boundary circles also gives $K$.\nThe Klein bottle cannot be embedded in $\\R^3$, since a closed surface embedded in $\\R^3$ bounds a region and is therefore orientable.\nIt has immersions in $\\R^3$ and embeddings in $\\R^4$.\n\n## Explanations\n\n### L1 human\n\nTake a rubber sheet, glue two opposite sides to make a tube, then glue the two ends of the tube with a flip.\nIn ordinary space the flip forces the tube to pass through its own wall, which gives the familiar glass bottle picture.\nThe result has no inside and no outside: a walker on it can come back mirror-reversed.\n",
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