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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *pseudocompact* if every [[continuous map|0VGFHTM75Y]] $f \\colon X \\to \\R$ is bounded, that is, there is a real number $M$ with $\\abs{f(x)} \\le M$ for all $x \\in X$.\n\n## Remarks\n\nExample: a [[compact space|0TVJZYB6SH]] $X$ is pseudocompact.\nThe image $f(X)$ of a continuous $f \\colon X \\to \\R$ is compact in $\\R$, so it is bounded.\n\nNon-example: the real line $\\R$ is not pseudocompact.\nThe identity map $f(x) = x$ is continuous and takes arbitrarily large values.\n\nSome authors require in addition that $X$ be a [[Tychonoff space|C5N2J2K6Z3]].\nA normal Hausdorff pseudocompact space is [[countably compact|S3N3CHZV89]].\n",
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