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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *compactly generated* if it satisfies the following condition.\nA subset $A \\subseteq X$ is closed whenever $u^{-1}(A)$ is closed in $K$ for every [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]] $K$ and every [[continuous map|0VGFHTM75Y]] $u \\colon K \\to X$.\n\n## Remarks\n\nIf $X$ is Hausdorff, the condition is equivalent to the classical one.\nA subset $A \\subseteq X$ is closed whenever $A \\cap C$ is closed in $C$ for every compact subset $C \\subseteq X$.\n\nExample: every [[metric space|0VKD1TP7F3]] is compactly generated.\nIf $A$ is not closed, there is a sequence $a_n \\in A$ converging to some $x \\notin A$.\nThe set $K = \\set{x} \\cup \\set{a_n : n \\ge 1}$ is compact, and $A \\cap K$ is not closed in $K$.\n\nNon-example: an uncountable set $X$ with the cocountable topology.\nIts compact subsets are finite, so every test map $u$ has finite image.\nFinite subsets of $X$ are closed and discrete, so every $u^{-1}(A)$ is closed.\nBut not every subset of $X$ is closed.\n",
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