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  "body": "\n## Summary\n\nA compactly generated space, or k-space, is a [[topological space|0CQHFPGJ45]] whose topology is determined by its compact pieces.\nA set is closed as soon as its trace on every compact piece is closed.\nMetric spaces and [[locally compact|0FE2SB8WVJ]] Hausdorff spaces are compactly generated.\nAlgebraic topology often works in the category of compactly generated spaces, because there products and function spaces behave well.\n\n## Explanations\n\n### L1 human\n\nTo check whether a set is closed, one may look at how it meets every compact part of the space.\nIn a compactly generated space, this local test is enough.\nFor example, in the plane a set is closed exactly when its intersection with every closed disc is closed.\nSo the topology of the plane is rebuilt from its compact discs.\n",
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