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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA *compactification* of $X$ is a pair $(K, e)$ with the following properties.\n\n1. $K$ is a [[compact space|0TVJZYB6SH]].\n2. $e \\colon X \\to K$ is a [[topological embedding|00ZG9Y9TR4]].\n3. The image $e(X)$ is [[dense|0BSJSX7ME2]] in $K$.\n\n## Remarks\n\nExample: $K = [0, 1]$ with the inclusion $e \\colon (0, 1) \\to [0, 1]$ is a compactification of $(0, 1)$.\nThe interval $[0, 1]$ is compact, the inclusion is an embedding for the subspace topology, and the closure of $(0, 1)$ is $[0, 1]$.\n\nNon-example: $K = [0, 2]$ with the inclusion of $(0, 1)$ fails clause 3.\nThe closure of $(0, 1)$ in $[0, 2]$ is $[0, 1]$, not $[0, 2]$.\n\nMany authors also require $K$ to be Hausdorff.\nTwo compactifications $(K, e)$ and $(K', e')$ are equivalent if there is a homeomorphism $h \\colon K \\to K'$ with $h \\circ e = e'$.\n",
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